MAT 300 Unit 3 Challenges Sophia MAT300 Unit 3 Challenges Sophia

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CHALLENEG 1 Which of these is NOT a possible outcome? a.)

Flipping tails on a coin b.)

Drawing a Queen of Hearts from a standard deck of cards c.)

Rolling an odd number that is less than 2 on a die d.)

Rolling a prime number that is larger than 5 on a die  If a die is 6 sided, then it has values corresponding from 1 to 6. So, a prime larger than 5 is not possible as the only number larger than 5 is 6 and 6 is not a prime number.

Which of these is NOT an outcome? a.)

Drawing a King of Diamonds from a standard deck of cards b.)

Rolling a 4 on a die c.)

Rolling an even number that is less than 2 on a die d.)

Flipping heads on a coin  If a die is 6 sided, then it has values corresponding from 1 to 6. An even number less than 2 is not possible as the only number less than 2 is 1 and 1 is not even. hich of these is NOT a possible outcome? a.)

Drawing an Ace of Clubs from a standard deck of cards b.)

Rolling a 6 on a die c.)

Drawing a Jack of Spades from a standard deck of cards d.)

Rolling a multiple of 8 on a die  Since a die is defined to be 6 sided with values corresponding from 1 to 6, an 8 is not possible.

A game involves a spinner that is evenly separated into four sections. To play the game, a player spins the spinner three times.

What is the number of individual outcomes when spinning the wheel three times? a.)

64 b.)

12 c.)

4 d.)

3  The fundamental counting principles for three events tells us the number of outcomes is n1*n2*n3, where n1, n2, and n3 are the number of outcomes at Event 1, Event 2, and Event 3 respectively. In this case, the possible number of outcomes in each event is 4, so n1 = n2 = n3 = 4. So the total number of outcomes is 4*4*4 = 64.

A game involves a spinner that is evenly separated into four sections, as well an eight-sided die. To play the game, each player spins the spinner once and rolls the die once.

What is the number of individual outcomes from spinning and rolling one time? a.)

12 b.)

32 c.)

8 d.)

4  The fundamental counting principles for two events tells us the number of outcomes is m*n, where m and n are the number of outcomes at Event 1 and 2 respectively. The first event is a spinner that has four sections or four outcomes, so m=4. The second event is an eight-sided die with eight outcomes, so n=8. The total number of outcomes is 4*8 = 32.

A game involves a two-sided coin, as well a six-sided die. To play the game, each player flips the coin once and rolls the die once.

What is the number of individual outcomes from flipping and rolling one time? a.)

2 b.)

6 c.)

8 d.)

12  The fundamental counting principles for two events tells us the number of outcomes is m*n, where m and n are the number of outcomes at Event 1 and 2 respectively. The first event is a two-sided coin that has two outcomes, so m=2. The second event is a six-sided die with six outcomes, so and n=6. The total number of outcomes is 2*6 = 12.

The theoretical probability of an event is the number of desired outcomes divided by all possible outcomes.

What is the probability of drawing an ace from a shuffled standard deck of playing cards? a.) b.) c.) d.)  Recall that there are 52 cards in a deck with a total of 4 aces in the deck. Since the draw is random, each outcome is equally likely and the probability of drawing an ace is: The theoretical probability of an event is the number of desired outcomes divided by all possible outcomes.

If the wheel is spun once, what is the probability of getting the number zero? a.) b.) c.) d.)  There are three zeros and a total of 12 possible spaces. Assuming the spin is done randomly, then the probability of getting a zero is: The theoretical probability of an event is the number of desired outcomes divided by all possible outcomes.

What is the probability of drawing a face card from a shuffled standard deck of playing cards? a.) b.) c.) d.) 

Recall that there are 52 cards in a deck with four suits (hearts, diamonds, clubs, and spades).

Each suit has three face cards of Jack, Queen, and King so there are 3*4 suits = 12 face cards.

Since the draw is random, each outcome is equally likely and the probability of drawing a face cards is: A fair die is rolled 25 times. It landed on the number 5 eight times.

What is the relative frequency of getting a 5 on the next roll? a.) b.) c.) d.)  If 25 rolls are representative of what we would expect on all rolls, then the probability of getting a 5 is: So there should be a 8/25 chance of getting a 5 on the next roll.

A box has 10 disks numbered 1 through 10. A disk is drawn at random 50 times with replacement. The disk numbered 7 was drawn 6 times.

What is the relative frequency of getting the disk numbered 7 on the next draw? a.) b.) c.) d.)  Since the draw is random and done with replacement, each draw has an equal likelihood from all other draws. If the 50 times it was drawn before is representative of all draws, the probability of drawing a 7 is: So there should be a 6/50 chance of a getting a disk numbered 7 on the next draw.

A coin is tossed 50 times and it lands on tails 20 times.

What is the relative frequency of getting a tails on the next toss? a.) b.) c.)

d.)  Assuming the 50 coin tosses is representative of all coin tosses, then the probability of getting a tail is: There would be a 2/5, or 40%, chance of a tail on the next toss.

Which of the following statements is true of a continuous probability distribution? a.)

The outcomes have a finite number of possibilities. b.)

The distribution can be described with a table. c.)

All outcomes can be listed. d.)

The outcomes can take any value within a given range.  If data is continuous, then inside of the range of possible values, the outcome can take on any value inside of that interval.

Which of the following statements is NOT true of a discrete probability distribution? a.)

You can describe the distribution with a table. b.)

You can list all of the outcomes. c.)

There are an infinite number of possible outcomes. d.)

There are a finite number of possible outcomes.  Since the data is discrete, there are only a finite and countable number of outcomes.

Which of the following is true of BOTH discrete and continuous probability distributions? a.)

The sum of all probabilities is less than or equal to 1. b.)

The sum of all probabilities is equal to 1. c.)

The sum of all probabilities lies between 0 to 1. d.)

The probability of each outcome is 1.  The rules for probability don't change with respect to the sum of all probability. So the total probability of a discrete or continuous distribution will always be 1. spinner is divided into four equal areas and colored red, green, yellow, and blue.

Which statement is most likely to be true? a.)

The probability of the spinner landing on the green region is close to 0.25 when the number of trials is less than 250. b.)

The probability of the spinner landing on the green region is close to 0.25 when the number of trials is more than 250. c.)

The probability of the spinner landing on the green region is equal to 0.25 when the number of trials is less than 25. d.)

The probability of the spinner landing on the green region is close to 0.25 when the number of trials is less than 25.  If each of the outcomes is equal, then the probability of any color is 1/4. If we spin the device a large enough amount of times, we would expect that each color would get closer to being about 1/4. Keep in mind that for smaller numbers of spins we might see colors that are not 1/4.

Which statement is most likely to be true upon rolling a standard die more than 100 times? a.)

The probability of getting a number less than 3 is equal to 0.33 and changes the probability of the next trial. b.)

The probability of getting a number less than 3 is close to 0.33 but does not change the probability of the next trial. c.)

The probability of getting a number less than 3 is close to 0.33 and changes the probability of the next trial. d.)

The probability of getting a number less than 3 is equal to 0.33 but does not change the probability of the next trial.  If a die is standard, then each outcome is equally likely. The total outcomes of a die are 1-6.

So the number of outcomes less than 3 are 1 or a 2. The probability of rolling one of these two number is 2/6, or 1/3 = 0.33. Since each trial is independent of all the other trials, we would expect a value less than 3, a third of the the time for a large enough number of rolls.

Which statement is most likely to be true for tossing two coins simultaneously? a.)

The probability of getting two heads is close to 0.25 when the number of trials is 5. b.)

The probability of getting two heads is close to 0.25 when the number of trials is 10. c.)

The probability of getting two heads is close to 0.25 when the number of trials is 25. d.)

The probability of getting two heads is close to 0.25 when the number of trials is 100.  If the coins are fair, in general we expect HH, TT, TH or HT roughly 1/4 of the time. On any individual toss we might observe any of these. However, when we toss more coins, we expect that the ratio would be 1/4, or 0.25, for each of these outcomes.

Which of the following statements is true? a.)

Using a sample size of 20 or less can be representative in simulations. b.)

Simulations utilize certain rules to get estimates closer to theoretical probability. c.)

To achieve theoretical certainty, trials of chance should be repeated. d.)

Simulations are based on trials of chance that mimic real-life events.  When simulating data, there is an element of randomness to the data that allows us to understand probabilities of real life events.

Correct Go to the Next Concept hich of the following statements is FALSE? a.)

Simulations enable you to approximate probability through experimental trials. b.)

Sample sizes of 10 or less are generally representative in simulations. c.)

All statements are false. d.)

Another name for simulations of probability is the Monte Carlo method.  When performing simulations, we generally repeat the simulation many times and use samples that are relatively large, unless they are trying to simulate high dimensional data.

Which of the following statements is true? a.)

Repeating trials of chance will enable you to achieve theoretical certainty. b.)

Sample sizes of 15 or less are generally representative in simulations. c.)

The Monte Carlo method is another name for simulations of probability. d.)

All statements are true.  Monte Carlo simulations are a good way to verify rules for probability by simulating data and then verifying the resulting probabilities.

Which of the following statements is true? a.)

Repeating trials of chance will enable you to achieve theoretical certainty. b.)

Sample sizes of 15 or less are generally representative in simulations. c.)

The Monte Carlo method is another name for simulations of probability. d.)

All statements are true.  Monte Carlo simulations are a good way to verify rules for probability by simulating data and then verifying the resulting probabilities.

Sarah plays a game in which she can purchase a ticket. Each ticket has several chances, or "catches," to win money.

The table below shows the probability of winning at each stage, and how much money the ticket can win at each catch. Every time Sarah plays the game, her ticket is played through each catch, which means she can win money at each stage.

Given the probabilities and payout values in this table, what is the expected value of Sarah's ticket? a.) $8.50 b.) $6.70 c.) $53.50 d.) $116.00  To get the expected values, we simply use the formula: ecca plays a game in which she can purchase a ticket. Each ticket has several chances, or "catches," to win money.

The table below shows the probability of winning at each stage, and how much money the ticket can win at each catch. Every time Becca plays the game, her ticket is played through each catch, which means she can win money at each stage.

Given the probabilities and payout values in this table, what is the expected value of Becca's ticket? a.) $1.20 b.) $4.60 c.) $41.00 d.) $10.25  To get the expected values, we simply use the formula: Zhi plays a game in which she can purchase a ticket. Each ticket has several chances, or "catches," to win money.

The table below shows the probability of winning at each stage, and how much money the ticket can win at each catch. Every time Zhi plays the game, her ticket is played through each catch, which means she can win money at each stage.

Given the probabilities and payout values in this table, what is the expected value of Zhi's ticket? a.) $95.00 b.) $30.00 c.) $0.95 d.) $0.70  To get the expected values, we simply use the formula: Correct Go to the Next Concept I need help with this question In a well-shuffled pack of cards, the odds in favor of picking a diamond card are .

What is the probability of drawing this card? a.) b.) c.) d.)  Recall that we can go from odds to a probability by rewriting it as the fraction .

So if the odds are , then the probability is: . he probability of rolling a 1 on a standard die is .

What are the odds in favor of getting a 1 upon rolling a die? a.) b.) c.) d.)  Recall that we can formally express odds as when we have a probability .

So if the probability is , then the odds are: .

The probability of randomly picking out an autobiography from a bookshelf is .

What are the odds in favor of choosing an autobiography? a.) b.) c.) d.)  Recall that we can formally express odds as when we have a probability .

So if the probability is then odds are: .

CHALLENEG 2 What is the probability of getting an odd number OR a 2 upon rolling a six-sided die? a.) b.) c.) d.)  On a six-sided die, the probability of getting an odd number {1, 3, 5} is: The probability of getting a 2 is: Since there is no overlap between rolling an odd number and rolling a 2, the probability of getting an odd number or a 2 is: What is the probability of getting a number less than 2 OR greater than 4 upon rolling a sixsided die? a.) b.) c.) d.) 

On a six-sided die, there is only one way to get a number less than 2, which is by rolling a 1.

So the probability of getting a number less than 2 is: There is only two numbers that are greater than 4 when rolling a six-sided die, which are 5 and 6. So the probability of getting a number greater than 4 is: Since there is no overlap between these two events, the probability of getting a number less than 2 OR a number greater than 4 is: What is the probability of getting a number less than 2 OR a prime number upon rolling a sixsided die? a.) b.) c.) d.)  On a six-sided die, there is only one way to get a number less than 2, which is by rolling a 1.

So the probability of getting a number less than 2 is: There are three primes numbers when rolling a six-sided die, which are {2, 3, 5}. So the probability of getting a prime number is: Since there is no overlap between these two events, the probability of getting a number less than 2 OR a prime number is: Correct hat is the probability of drawing a diamond from a standard deck of 52 cards AND getting tails upon flipping a fair coin? a.) b.) c.) d.)  If we let event = drawing a diamond and event = getting tails, then we may note since there are 13 diamonds in a standard deck of 52 cards, and since there is a 50% chance of getting tails when flipping a coin.

Since these events are independent (meaning the outcome of event A does not impact the outcome of event B), we can note that:

The probability of drawing a blue ball from Bag A is 1/5 and the probability of drawing a red ball from Bag B is 2/3.

What is the probability of drawing a blue ball from Bag A AND a red ball from Bag B? a.) b.) c.) d.)  If we let event = drawing blue ball from Bag A and event = drawing a red ball from Bag B, then we may note and .

Since these events are independent (meaning the outcome of event A does not impact the outcome of event B), we can note that: A bag contains 3 red balls and 5 green balls.

What is the probability of picking a red ball from the bag AND getting heads upon flipping a fair coin? a.) b.) c.) d.)  If we let event = drawing a red ball and event = getting heads, then we may note since there are 3 red balls out of a total of 8 balls, and since there is a 50% chance of getting heads when flipping a coin.

Since these events are independent (meaning the outcome of event A does not impact the outcome of event B), we can note that: Ashley noticed that a black number had appeared five times in a row on the following roulette table. She now felt very strongly that a red number would be next.

On a single spin, the probability of getting a red number is .

Considering the last five spins resulted in black numbers, what is the probability of the next spin being red? a.) because each spin is dependent.

b.) because each spin is independent. c.) because each spin is dependent. d.) because each spin is independent.  Because we can assume that the spins are independent, which implies that one spin does not affect the likelihood of a future spin, the probability of red should still be . hi noticed that an odd number had appeared eight times in a row on the following roulette table. She now felt very strongly that an even number would be next.

On a single spin, the probability of getting an even number is .

Considering the last eight spins resulted in odd numbers, what is the probability of getting an even number on the next spin? a.) because each spin is independent. b.) because each spin is independent. c.) because each spin is dependent. d.) because each spin is dependent.  Because we can assume that the spins are independent, which implies that one spin does not affect the likelihood of a future spin, the probability of even should still be .

Greg noticed that a green space had appeared three times in a row on the following roulette table. He now felt very strongly that another green space would be next.

On a single spin, the probability of getting a green space is .

Considering the last three spins resulted in a green space, what is the probability of getting a green space on the next spin? a.) because each spin is dependent. b.) because each spin is independent. c.) because each spin is independent. d.) because each spin is dependent.  Because we can assume that the spins are independent, which implies that one spin does not affect the likelihood of a future spin, the probability of green should still be .

Correct Go to the Next Concept 

    What is the probability that a card chosen is orange OR striped? a.) b.) c.) d.)  There are 8 cards overall. There are 3 orange cards and 2 blue cards that are striped with no overlap. So the total cards that are orange OR striped is 5 cards. This tells us the probability that a card chose is orange OR striped is:      What is the probability that a card chosen is blue OR has polka dots? a.) b.) c.) d.)  There are 8 cards overall. There are 5 blue cards and 1 orange card with polka dots with no overlap. So the total cards that are blue OR polka dot is 6 cards. This tells us the probability that a card chosen is blue OR has polka dots is:

What is the probability that a card is striped AND blue? a.) b.) c.) d.)  There are 8 cards overall. There are 2 cards that are striped AND blue. This tells us the probability that a card is striped AND blue is: Two sets A and B are shown in the Venn diagram below.

Which statement is true? a.)

Set A has 56 elements. b.)

There are a total of 120 elements shown in the Venn diagram. c.)

Sets A and B have 10 common elements. d.)

Set B has 42 elements.  The overlap, or intersection, shows the elements that the two sets share, which is 10.

Sets P and Q are shown in the Venn diagram below.

Which statement is true? a.)

Set Q has 38 elements. b.)

Set P has 17 elements. c.)

Sets P and Q have 30 common elements. d.)

There are a total of 25 elements shown in the Venn diagram.  To identify the elements in Set P, we need to consider the whole circle, or 15 + 2 = 17 elements.

Two sets --M and N-- are shown in the Venn diagram below.

Which statement is FALSE?

a.)

Set M has 30 elements. b.)

Sets M and N have 4 common elements. c.)

There are a total of 112 elements shown in the Venn diagram. d.)

Set N has 22 elements.  If we assume all values in the Venn diagram represent the total number of elements, we simply sum up the 4 parts to get 26 + 4 + 18 + 60 = 108 elements overall.

Which pair of events are overlapping when rolling a single six-sided die? a.)

Getting a 2 Getting an odd number b.)

Getting a prime number Getting a number greater than 5 c.)

Getting an even number Getting a number less than 5 d.)

Getting an even number Getting a number less than 2  In a six-sided die, the even numbers are {2, 4, 6}. Getting a number less than 5 can be {1, 2, 3, 4}. So the overlap between these groups, also called the intersection, is {2, 4}. hich pair of events are NOT overlapping when rolling a single die? a.)

Getting a number greater than 5 Getting an odd number b.)

Getting a number less than 5 Getting a 2 c.)

Getting a number greater than 3 Getting a 6 d.)

Getting a number less than 5 Getting a prime number  In a six-sided die, the only value greater than 5 is {6}. Getting an odd number can be {1, 3, 5}. Since these events do not have any numbers in common, these are non-overlapping events.

Correct Go to the Next Concept Which pair of events are NOT overlapping when rolling a single die? a.)

Getting a number greater than 2

Getting a number less than 5 b.)

Getting an even number Getting a prime number c.)

Getting a number greater than 4 Getting a number less than 2 d.)

Getting an odd number Getting a prime number  In a six-sided die, the values greater than 4 is {5, 6}. Getting a number less than 2 can only be {1}. Since these events do not have any numbers in common, these are non-overlapping events.

A roulette wheel has 38 slots total, 36 of which are numbered 1 through 36 and 2 green slots labeled "0" and "00."

For any spin of the wheel, what is the probability of the roulette ball NOT landing on green? a.) b.) c.) d.)  Using the complemental rule, we can note the probability of NOT landing on green is equal to: roulette wheel has 38 slots total, 36 of which are numbered 1 through 36 and 2 green slots labeled "0" and "00."

For any spin of the wheel, what is the probability of NOT getting a number 1 through 18? a.) b.) c.) d.) 

Using the complemental rule, we can note the probability of NOT getting a number 1 through 18 is equal to: A roulette wheel has 38 slots total, 36 of which are numbered 1 through 36 and 2 green slots labeled "0" and "00."

For any spin of the wheel, what is the probability of the roulette ball NOT landing on red? a.) b.) c.) d.)  Using the complemental rule, we can note the probability of NOT landing on red is equal to: Correct Go to the Next Concept In a well-shuffled, standard 52 card deck, what is the probability that a card is a Queen, given that it is a black card? Answer choices are in a percentage format, rounded to the nearest whole number. a.)

33% b.)

25% c.)

8% d.)

10%  The probability that a card is a Queen, given that it is a black card, can be expressed as the conditional probability: The probability that a card is both a Queen AND a black card is , as there are only 2 cards in a deck that have both qualities.

The probability of a black card is as half of the deck of cards are black.

So the probability of a card being a Queen given that it is a black card is: .

CHALLENEG 3

In a well-shuffled, standard 52 card deck, what is the probability that a card is a Two, given that it is NOT a face card? Answer choices are in a percentage format, rounded to the nearest whole number. a.)

10% b.)

8% c.)

4% d.)

29%  The probability that a card is a Two given that it is NOT a face card can be expressed as the conditional probability: The probability that a card is both a Two AND not a face card is , as there are only 4 cards in a deck that have both qualities.

The probability that a card is NOT a face card is as there are 12 face cards in a deck of 52 cards so the remaining 40 are non-face cards.

So the probability of a card being a Two given that it is a NOT a face card is: .

In a well-shuffled, standard 52 card deck, what is the probability of a card being a Jack, given that it is a face card? Answer choices are in a percentage format, rounded to the nearest whole number. a.)

33% b.)

8% c.)

25% d.)

23%  The probability that a card is a Jack, given that it is a face card, can be expressed as the conditional probability: The probability that a card is both a Jack AND a face card is , as there are only 4 cards in a deck that have both qualities.

The probability of a face card is as there are a total of 12 face cards in a deck of cards.

So the probability of a card being a Jack given that it is a face card is: .

A total of 70 people were surveyed. The following Venn diagram shows the number of people who like pizza and who like soda

What is the probability that someone likes pizza, assuming that they also like soda? Answer choices are in a percentage format, rounded to the nearest whole number. a.)

7% b.)

23% c.)

29% d.)

31%  This is an example of a conditional event, since we are assuming they like soda. To get the probability of people who like pizza, given that they also like soda, we can use the following formula: .

A total of 55 students were surveyed. The following Venn diagram shows the number of students who like math and who like art.

What is the probability that someone likes art, assuming that they also like math? Answer choices are in a percentage format, rounded to the nearest whole number. a.)

33% b.)

50% c.)

11% d.)

75%  This is an example of a conditional event, since we are assuming they like math. To get the probability that someone likes art, given that they also like math, we can use the following formula: A total of 80 people were surveyed. The following Venn diagram shows the number of people who like cats and who like dogs.

What is the probability that someone likes dogs, assuming that they also like cats? Answer choices are in a percentage format, rounded to the nearest whole number. a.)

41% b.)

51% c.)

21% d.)

71%  This is an example of a conditional event, since we are assuming they like cats. To get the probability that someone likes dogs, given that they also like cats, we can use the following formula: A school cafeteria wants to find out what combinations of food and beverage students are ordering. The results are represented in the two-way table below:

Of the students who ordered pizza, what is the probability that a student also ordered a soda? a.) b.) c.) d.)  Since we are focusing just on the population of students who ordered pizza, we can isolate our view to just pizza. There were a total of 11 + 18 + 28, or 57, students who ordered pizza.

Of those 57 students who ordered pizza, there were 28 who ordered soda, or a probability of .

A school cafeteria wants to find out what combinations of food and beverage students are ordering. The results are represented in the two-way table below:

Of all the students, what is the probability that a student ordered a hamburger and a slurpee? a.) b.) c.) d.)  Since we are focusing on all the students, we need to find the total number. To find the total, we can add up ALL the numbers in the contingency table: 11 + 18 + 28 + 7 + 19 + 12 + 23 + 5 + 14 = 137 students.

Of those 137 students, there were 19 students who ordered a hamburger and a slurpee, or a probability of .

A school cafeteria wants to find out what combinations of food and beverage students are ordering. The results are represented in the two-way table below:

Of the students who ordered salad, what is the probability that a student also ordered water? a.) b.) c.) d.)  Since we are focusing just on the population of students who ordered salad, we can isolate our view to just salad. There were a total of 23 + 5 + 14, or 42, students who ordered salad.

Of those 42 students who ordered salad, there were 23 that ordered water, or a probability of .

The following contingency table shows the number of games won and lost at two locations where a local baseball team had played.

What is the probability that the team won a game if it played at Lincoln Park? Answer choices are rounded to the hundredths place. a.)

0.70 b.)

0.54 c.)

0.65 d.)

0.35  This is an example of a conditional event, since we are looking at those who played at Lincoln Park. Keep in mind that there were a total of 6 + 7 + 4 + 3 = 20 games played.

To get the probability that a team won, given it played at Lincoln Park, we can use the formula: There were 7 out of 20 games that were won and played at Lincoln Park and 7 + 3, or 10, out of 20 games played at Lincoln Park.

Correct Go to the Next Concept he following contingency table shows the number of games won and lost at two locations where a local baseball team had played.

What is the probability that the team lost a game if it played at Lincoln Park? Answer choices are rounded to the hundredths place. a.)

0.15 b.)

0.30 c.)

0.43 d.)

0.40  This is an example of a conditional event, since we are looking at those who played at Lincoln Park. Keep in mind that there were a total of 6 + 7 + 4 + 3 = 20 games played.

To get the probability that a team lost, given it played at Lincoln Park, we can use the formula: There were 3 out of 20 games that were lost and played at Lincoln Park and 7 + 3, or 10, out of 20 games played at Lincoln Park.

The following contingency table shows the number of games won and lost at two locations where a local baseball team had played.

What is the probability that the team won a game if it played at Daryl Fields? Answer choices are rounded to the hundredths place. a.)

0.30 b.)

0.60 c.)

0.50 d.)

0.85  This is an example of a conditional event, since we are looking at those who played at Lincoln Park. Keep in mind that there were a total of 6 + 7 + 4 + 3 = 20 games played.

To get the probability that a team won, given it played at Daryl Fields, we can use the formula: There were 6 out of 20 games that were won and played at Daryl Fields and 6 + 4, or 10, out of 20 games played at Daryl Fields.

In a standard deck of cards, what is the probability of drawing two face cards in a row?

Answer choices are in the form of a percentage, rounded to the nearest whole number. a.)

28% b.)

71% c.)

5% d.)

10%  These events are dependent because the probability of drawing a second card will be impacted based on the outcome of the first card. We must use the general rule of multiplication for getting this probability.

If you want the probability of drawing two face cards in a row, this is the same as the conditional probability of drawing a face card second, given that we drew a face card first.

We can use the following formula: The probability of drawing a face card on the first draw would be .

The conditional probability that the second card is a face given that the first card is a face card would be because there are now only 11 face cards remaining, and only 51 total cards since the first card has been drawn.

In a standard deck of cards, what is the probability of drawing a face card followed by drawing a non-face card? Answer choices are in the form of a percentage, rounded to the nearest whole number. a.)

27% b.)

18% c.)

4% d.)

45%  These events are dependent because the probability of drawing a second card will be impacted based on the outcome of the first card. We must use the general rule of multiplication for getting this probability.

If you want the probability of drawing a face card followed by a non-face, this is the same as the conditional probability of drawing a non-face second, given that we drew a face card first.

We can use the following formula:

The probability of drawing a face card on the first draw would be .

The conditional probability that the second card is a non-face given that the first card is a face card would be because there would still be 40 non-face cards remaining, but there are now only 51 total cards remaining since the first card has been drawn. n a standard deck of cards, what is the probability of drawing a black card followed by drawing a heart? Answer choices are in the form of a percentage, rounded to the nearest whole number. a.)

10% b.)

13% c.)

26% d.)

75%  These events are dependent because the probability of drawing a second card will be impacted based on the outcome of the first card. We must use the general rule of multiplication for getting this probability.

If you want the probability of drawing a black card followed by a heart, this is the same as the conditional probability of drawing a heart second, given that we drew a black card first. We can use the following formula: The probability of drawing a black card on the first draw would be .

The conditional probability that the second card is a heart given that the first card is a black card would be because there would still be 13 hearts remaining, but there are now only 51 total cards since the first card has been drawn.

In a standard deck of cards, what is the probability of drawing a red card OR a face card?

Answer choices are in the form of a percentage, rounded to the nearest whole number. a.)

23% b.)

62% c.)

73% d.)

25%  Here we have overlapping events because a card can be both a red card and a face card.

To find the probability of drawing a red OR a face card, we can use the following formula:

There are 26 red cards out of 52 cards, 12 face cards out of 52 cards, and 6 out of 52 cards that are both red and face. So we can say: n a standard deck of cards, what is the probability of drawing an Ace OR a black card?

Answer choices are in the form of a percentage, rounded to the nearest whole number. a.)

33% b.)

54% c.)

50% d.)

25%  Here we have overlapping events because a card can be both an Ace or a black card.

To find the probability of drawing an Ace OR a black card, we can use the following formula: There are 4 Aces out of 52 cards, 26 black cards out of 52 cards, and 2 out of 52 cards that are both Ace and black. So we can say: In a standard deck of cards, what is the probability of drawing a diamond OR a face card?

Answer choices are in the form of a percentage, rounded to the nearest whole number. a.)

17% b.)

50% c.)

25% d.)

42%  Here we have overlapping events because a card can be both a diamond and a face card.

To find the probability of drawing a diamond OR a face card, we can use the following formula: There are 13 out of 52 diamonds, 12 out of 52 face cards, and 3 out of 52 cards that are both face and diamond. So we can say:   

  Using this Venn diagram, the probability that event A or event B occurs is __________. a.)

0.76 b.)

0.08 c.)

0.68 d.)

0.84  Here we have overlapping events so we can note that: The probability that A happens is the whole circle of A, or .

The probability that B happens is the whole circle of B, or .

The probability that A and B happens is the overlap of the Venn diagram, or

You can also note that if you simply add up all the parts, 0.18 + 0.08 + 0.58, you can also get the final result.      Using this Venn diagram, the probability that event A or event B occurs is __________. a.)

0.57 b.)

0.45 c.)

0.31 d.)

0.69  Here we have overlapping events so we can note that:

The probability that A happens is the whole circle of A, or .

The probability that B happens is the whole circle of B, or .

The probability that A and B happens is the overlap of the Venn diagram, or .

You can also note that if you simply add up all the parts, 0.30 + 0.12 + 0.27, you can also get the final result.      Using this Venn diagram, the probability that event A or event B occurs is __________. a.)

0.55 b.)

0.85 c.)

0.70 d.)

0.15  Here we have overlapping events so we can note that: The probability that A happens is the whole circle of A, or .

The probability that B happens is the whole circle of B, or .

The probability that A and B happens is the overlap of the Venn diagram, or .

You can also note that if you simply add up all the parts, 0.25 + 0.15 + 0.45, you can also get the final result. n a factory that manufactures bolts, the first machine manufactures 70% of the bolts and a second machine manufactures the remaining 20%. The percentage of defective bolts is 3% and 5%, respectively.

An employee picks a bolt off a shelf at random and it is from the first machine (event A).

If you want to know the probability that it is a defective bolt (event C), which formula would you use? a.) b.) c.) d.)  Bayes' rule is a mathematical theorem that allows you to amend probability statements based on new information. With this rule, you can turn around the conditional probability statement and find the probability of the first event happening given the second event happened by using the probability of the second event happening given the first event happened.

In this case, the first event is that the bolt is defective (event C). The second event is that it was picked from the first machine (event A). Therefore, we can say the conditional probability that a bolt is defective, given it was from the first machine, or P(C|A), is this formula.

A car producer stocks three types of tires: A, B, and C. Let P(A) = 0.40, P(B) = 0.15 and P(C) = 0.45. The percentage of defective tires is 2%, 1% and 5%, respectively.

Someone picks a tire off the shelf at random and it is Brand A. If you want to know the probability that it is a defective tire (event D), which formula would you use? a.) b.) c.) d.)  Bayes' rule is a mathematical theorem that allows you to amend probability statements based on new information. With this rule, you can turn around the conditional probability statement and find the probability of the first event happening given the second event happened by using the probability of the second event happening given the first event happened.

In this case, the first event is that the tire is defective (event D). The second event is that it was Brand A (event A). Therefore, we can say the conditional probability that a tire is defective, given it was Brand A, or P(D|A), is this formula.

Correct On any given day in which it had rained, there was a 40% chance that the morning was cloudy.

When you wake up in the morning, you notice that it is cloudy (event C).

If you want to know the probability that it will rain (event R), which formula would you use? a.) b.) c.) d.)  Bayes' rule is a mathematical theorem that allows you to amend probability statements based on new information. With this rule, you can turn around the conditional probability statement and find the probability of the first event happening given that the second event happened by using the probability of the second event happening given the first event happened.

In this case, the first event is that it rained (event R). The second event is that it was cloudy in the morning (event C). Therefore, we can say the conditional probability that it will rain, given it was cloudy in the morning, or P(R|C), is this formula.

Correct Go to the Next Concept elect the statement that indicates a "false negative." a.)

A pregnancy test tells a woman she is not pregnant when she is. b.)

Sending an innocent man to jail. c.)

Test results indicate that a patient has cancer when he does not. d.)

A control strip indicates a two degree increase in temperature in the storage area when, in fact, there was no change in temperature.  Recall, a false negative result is when the thing being tested for is mistakenly shown to be absent and in fact it's present. So, even though the pregnancy test being tested mistakenly shows that the woman was not pregnant, she in fact was pregnant.

Select the statement that indicates a "false negative." a.)

A temperature strip indicates a change in temperature when one did not occur. b.)

The store's alarm sounds off even though a break-in did not happen. c.)

A customer gets strep throat after drinking milk that passed inspection for streptococcus. d.)

An innocent man is sent to jail.  Recall, a false negative result is when the thing being tested for is mistakenly shown to be absent and in fact it's present. So a customer getting strep from a milk bottle that had passed

infection and that streptococcus was not present, shows that is shouldn't have passed inspection, indicating a false negative.

Select the statement that indicates a "false positive." a.)

A pregnancy test tells a woman she is not pregnant when, in fact, she is. b.)

A jury concludes that a man is innocent when he is actually guilty. c.)

A control strip fails to indicate a two degree increase in temperature in the storage area. d.)

Testing a shipment of dairy products reveals salmonella when, in fact, the bacteria is not present.  A false positive result is when the thing being tested for is mistakenly shown to be present when in fact it's not present. So, even though the test indicates a case of salmonella, the bacteria is not actually present.

A researcher was interested in how many people finish the Sunday versus the Monday crossword puzzle. The results of the research show the following: Which of the following explains how this is an example of Simpson's paradox? a.)

Finish rates were higher for the Monday puzzle overall, but the finish rates for females is lower on Monday. b.)

The finish rates are higher for the Sunday puzzle than the Monday puzzle for both males and females. c.)

Finish rates were higher for the Monday puzzle for both men and women, but overall finish rates are higher for the Sunday puzzle. d.)

Finish rates were higher for males overall, but the finish rates for females are higher for the Sunday puzzle.  Simpson's paradox shows us how a trend can be reversed from the true overall trend when subgroups are examined. We note that overall, on Sunday people generally finish the cartoons more than on Monday (27.6% compared to 24.1%).

However, the subgroups Male and Female have higher rates on Monday than on Sunday.

Females finish the Monday puzzle faster than the Sunday puzzle (21.8% compared to 20.3%.

Males finish the Monday puzzle faster than the Sunday puzzle (77.8% compared to 46.5%).

So the trends are reversed. researcher was interested in baseball results for the Minnesota Twins and Chicago Cubs when they played at home and away. The results of the research show the following: Which of the following explains how this is an example of Simpson's paradox? a.)

Overall win rates are lower for the Minnesota Twins than the Chicago Cubs. b.)

Win rates were higher for the Chicago Cubs for both home and away games, but overall win rates are higher for the Minnesota Twins. c.)

Win rates were higher for the Chicago Cubs overall, but lower for home games. d.)

Win rates were higher for away games overall but lower than home for the Minnesota Twins.  Simpson's paradox shows us how a trend can be reversed from the true overall trend when subgroups are examined. We note that the Cubs have higher win rates in both Home games and Away games compared to the Minnesota Twins (Home: 14.7% compared to 13.8%; Away: 49% compared to 48%). However, overall the Twins have a higher rate of winning than the Cubs (29.1% compared to 28.3%). So the trends are reversed. esearcher was interested in how many male and females were accepted into a university's graduate program. The results of the research show the following: Which of the following explains how this is an example of Simpson's paradox? a.)

Overall acceptance rates are lower for the Engineering graduate program than the Psychology graduate program. b.)

Acceptance rates were higher for the Psychology graduate program for both men and women, but overall acceptance rates are higher for the Engineering graduate program. c.)

Acceptance rates were higher overall for males but lower than females for the Engineering graduate program. d.)

Acceptance rates were higher for the Psychology graduate program overall, but lower for females.  Simpson's paradox shows us how a trend can be reversed from the true overall trend when subgroups are examined. We note that overall acceptance rates are higher for engineering majors than psychology majors (28.0% compared to 25.6%). However, in subgroups by gender, we can see that for both males and females, psychology has higher acceptance rates versus engineering majors (Males: 46.7% compared to 71.4%; Females: 20.7% compared to 23.4%). So the trends are reversed.

Which of the following is NOT an example of a binomial distribution? a.)

The probability of a dart player missing the bullseye seven times in eight attempts. b.)

The probability of rolling a die eight times and getting an odd number or a prime number four times. c.)

The probability of flipping a coin four times and never landing on tails. d.)

The probability of a basketball player making a free throw seven times in ten attempts.  When exploring a binomial, there can only be one variable we look at with two possible outcomes. In this case, we are exploring an odd number or a prime, which are two variables with more than two possible outcome

Which of the following is NOT an example of a binomial distribution? a.)

The probability of rolling a die three times and never rolling a 5. b.)

The probability of drawing a card and getting a red or face card. c.)

The probability of a basketball player making a free throw six times in eight attempts. d.)

The probability of a dart player hitting the bullseye three times in eight attempts.  When exploring a binomial, there can only be one variable we look at with two possible outcomes. In this case, we are exploring a red card or a face card, which are two variables with more than two possible outcome Which of the following is NOT an example of a binomial distribution? a.)

The probability of a basketball player missing a free throw six times in nine attempts. b.)

The probability of an archer hitting the bullseye four times in seven attempts. c.)

The probability of rolling a die five times and getting an even number or a prime number three times. d.)

The probability of flipping a coin four times and never landing on heads.  When exploring a binomial distribution, there can only be one variable we look at with two possible outcomes. In this case, we are exploring an even number or a prime, which is considered two different variables with more than two possible outcomes. basketball player has a 25% accuracy rate at making three-point shots. Mark thought it was reasonable that each attempt was independent and the probability stayed at 25% for this player.

Using the geometric distribution formula, what is the probability that the basketball player makes his first three-point shot on the third attempt? Answer choices are rounded to the hundredths place. a.)

0.14 b.)

0.25 c.)

0.38 d.)

0.56  In this case, a "success" will be considered making a three-point shot. The basketball player has a 25% chance to make the shot, or 0.25. We are also looking for the probability that the first success occurs on the third trial. So we have:

We can use the following formula: A basketball player has a 70% accuracy rate for making free throws. Mark thought that each attempt was independent and the probability stayed at 70% for this player.

During a game, this player was fouled and given the chance to take two free throws.

Using the geometric distribution formula, what is the probability that this player misses his first free throw, but makes the second one? Answer choices are rounded to the hundredths place. a.)

0.50 b.)

0.30 c.)

0.21 d.)

0.70  In this case, a "success" will be considered making a free throw. The basketball player has a 70% chance to make the shot, or 0.70. We are also looking for the probability that the the first success occurs on the second trial. So we have:

We can use the following formula: A basketball player has a 60% accuracy rate for making free throws. During practice, this player fails to make a free throw three times in a row, but is finally successful on the fourth attempt.

Using the geometric distribution formula, what is the probability of this player successfully making a free throw on the fourth attempt? Each attempt is independent of each other and answer choices are rounded to the hundredths place. a.)

0.10 b.)

0.40 c.)

0.60 d.)

0.04 

In this case, a "success" will be considered making a free throw. The basketball player has a 60% chance to make the shot, or 0.60. We are also looking for the probability that the first success occurs on the fourth trial. So we have:

We can use the following formula: A basketball player has a 60% accuracy rate for making free throws. During practice, this player fails to make a free throw three times in a row, but is finally successful on the fourth attempt.

Using the geometric distribution formula, what is the probability of this player successfully making a free throw on the fourth attempt? Each attempt is independent of each other and answer choices are rounded to the hundredths place. a.)

0.10 b.)

0.40 c.)

0.60 d.)

0.04  In this case, a "success" will be considered making a free throw. The basketball player has a 60% chance to make the shot, or 0.60. We are also looking for the probability that the first success occurs on the fourth trial. So we have:

We can use the following formula: The average number of road accidents that occur on a particular stretch of road during a year is 11.

Using the Poisson distribution formula, what is the probability of observing exactly 7 accidents on this stretch of road next year? Answer choices are rounded to the hundredths place. a.)

0.06 b.)

0.01 c.)

0.02 d.)

0.14  We can use the Poisson distribution to find the probability of observing exactly 7 accidents when the average number is 11. In this case: So we have: A small insurance company has determined that on average it receives 6 property damage claims per day.

What is the probability that the company will receive 7 property damage claims on a randomly selected day? Answer choices are rounded to the hundredths place. a.)

0.44 b.)

0.34 c.)

0.14 d.)

0.59  We can use the Poisson distribution to find the probability of observing exactly 7 property damage claims when the average number is 6. In this case:

So we have: pproximately 54% of mathematics students do their homework on time.

In a class of 250 students, what is the mean, variance, and standard deviation if we assume normality and use the normal distribution as an approximation of the binomial distribution?

Answer choices rounded to the nearest whole number. a.)

Mean = 135 Variance = 8 Standard Deviation = 62 b.)

Mean = 135 Variance = 62 Standard Deviation = 8 c.)

Mean = 53 Variance = 62 Standard Deviation = 8

d.)

Mean = 54 Variance = 8 Standard Deviation = 62  If is the number of trials and and are the probabilities of success and failure, then the normal approximation has a mean of , a variance of , and a standard deviation of .

In this case, , , and .

Using the normal approximation to the binomial, we can note that: Approximately 60% of mathematics students do their homework on time.

In a class of 100 students, what is the mean, variance, and standard deviation if we assume normality and use the normal distribution as an approximation of the binomial distribution?

Answer choices rounded to the nearest whole number. a.)

Mean = 60 Variance = 24 Standard Deviation = 5 b.)

Mean = 40 Variance = 5 Standard Deviation = 24 c.)

Mean = 40 Variance = 24 Standard Deviation = 5 d.)

Mean = 60 Variance = 5 Standard Deviation = 24  If is the number of trials and and are the probabilities of success and failure, then the normal approximation has a mean of , a variance of , and a standard deviation of .

In this case, , , and .

Using the normal approximation to the binomial, we can note that: Approximately 40% of students enjoy basketball.

In a class of 400 students, what is the mean, variance, and standard deviation if we assume normality and use the normal distribution as an approximation of the binomial distribution?

Answer choices rounded to the nearest whole number. a.)

Mean = 160 Variance = 10 Standard Deviation = 96

b.)

Mean = 240 Variance = 10 Standard Deviation = 96 c.)

Mean = 160 Variance = 96 Standard Deviation = 10 d.)

Mean = 240 Variance = 96 Standard Deviation = 10  If is the number of trials and and are the probabilities of success and failure, then the normal approximation has a mean of , a variance of , and a standard deviation of .

In this case, , , and .

Using the normal approximation to the binomial, we can note that: Approximately 40% of students enjoy basketball.

In a class of 400 students, what is the mean, variance, and standard deviation if we assume normality and use the normal distribution as an approximation of the binomial distribution?

Answer choices rounded to the nearest whole number. a.)

Mean = 160 Variance = 10 Standard Deviation = 96 b.)

Mean = 240 Variance = 10 Standard Deviation = 96 c.)

Mean = 160 Variance = 96 Standard Deviation = 10 d.)

Mean = 240 Variance = 96 Standard Deviation = 10  If is the number of trials and and are the probabilities of success and failure, then the normal approximation has a mean of , a variance of , and a standard deviation of .

In this case, , , and .

Using the normal approximation to the binomial, we can note that: