PROBABILITY & COUNTING RULES

CHAPTER 2: PROBABILITY & COUNTING RULES 2.1 SAMPLE SPACE AND PROBABILITY Experiment

: is a process of obtaining an outcome.

Outcome

: the result of a single trial of a probability experiment.

Sample space : the set of all possible outcomes resulting from an experiment. Sample space denoted as S. Event

: consist of a set of outcomes of a probability experiment and it is taken from a sample space. An event is a subset of a sample space.

Outcomes

Sample Space

Pass, Fail

S={Pass, Fail}

Select a student

Male , Female

S={Male,Female}

Toss a coin twice

HH,HT,TH,TT

S={HH,HT,TH,TT}

1,2,3,4,5,6

S={1,2,3,4,5,6}

Experiment Take an examination

Roll a dice once

2.2 COUNTING RULES If there are m ways of doing one thing and n ways of doing another thing, therefore they will be m x n ways of doing both.

Example 1: How many sample points are there in a sample space when a pair of dice is thrown once?

Solution:

n1 = 6 n2 = 6 Total no. of ways = n1 x n 2 = 6 x 6 = 36

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CHAPTER 2: PROBABILITY & COUNTING RULES Example 2: A car dealer offer types of cars for sale with brand Chevrolet, Mustang, Cardillac, Cryst with fuel consumption of brand Shell, Esso and Caltex. In how many ways can he sell the cars?

Solution:

n1 = 4 (type of cars) n 2 = 3 (fuel brands) Total no. of ways = n1 x n 2 = 4 x 3 = 12 ways 1. You want to choose president and vice president. There are 3 persons, Ali, Ahmad and Adam. How many ways for choose them?

2. There are 4 ways for journey from town A to town B and 3 ways from town B to town C. how many ways from town A to town C?

3. Given 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. How many numbers with the four digits (not start with zero) can be form if: a. Repetition is allowed.

b. Repetition is not allowed.

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CHAPTER 2: PROBABILITY & COUNTING RULES 4. How many ways the word SABUN can be arranged which start with N?

2.2.1 PERMUTATIONS ~ is an arrangement of things in a particular order.

CASE 1: We are given n different items; we want to arrange all the n items.

No. of arrangements = n!

Example: No. of permutation of 3 words ABC is 3! = 6 Example: 1. In how many ways can a committee consisting of a chairman, vice chairmen, a secretary and a treasurer be formed from 4 nominated members?

2. In how many ways can five letters A, B, C, D and E be arranged?

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CHAPTER 2: PROBABILITY & COUNTING RULES CASE 2: A permutation of n elements taken r at a time is an ordered arrangement and without repetition We want to arrange r out of n items. (r ≤ n ) No. of arrangements, n Pr =

n! (n − r )!

Note: the above formula can only be used for arrangement in a line and without repetition.

Example: 1. Find the number of different arrangements to form 4 letters from A, B, C, D, E, F and G.

2. How many different number plates can be formed if each contains 3 letters A, C, E followed by 3 digits 4, 7, and 8.

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CHAPTER 2: PROBABILITY & COUNTING RULES CASE 3: There are some similar items in n items given.

No. of arrangements =

n! n1!n2 !......nr !

where n1!n2 !......nr ! =number of items which are alike.

Example: 1. In how many ways can the alphabet in the name ‘LIZANA’ can be arranged?

2. STATISTICS

3. CORPORATION

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CHAPTER 2: PROBABILITY & COUNTING RULES CASE 4: Circular permutation: permutation that occurs by arranging objects in a circle.

The number of permutations of n objects arranged in a circle is (n-1)!

Example: In how many ways can 7 people be arranged in a circular table?

Solution: Total no. of ways = (7-1)! = 6! = 720

2.2.2 COMBINATIONS ~ Combination is the selection of one or more items from a collection of items where order does not important. ~ The number of combination of n distinct objects taken r at a time is: No. of combinations,

n

Cr =

n! r!(n − r )!

Example: 1. How many five different questions can be answered from 10 questions?

2. Find the number of combinations consisting of three letters that can be formed from the letters A, B, C, D, E, and F.

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CHAPTER 2: PROBABILITY & COUNTING RULES 3. A team of 4 is chosen at random from a group of 5 girls and 6 boys. In how many ways can the team be chosen if: a. There are no restrictions.

b. There must be more boys than girls in the team.

4. Find the number of committees that can be formed from 8 men and 4 women if the committee of 5 persons must consist of a. Exactly 5 persons

b. 3 men and 2 women

c. At least 3 men

d. At most 3 women

5. A team of 4 is chosen at random from a group of 5 girls and 6 boys. Find the probability that the team be chosen if there must be more boys than girls in the team.

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CHAPTER 2: PROBABILITY & COUNTING RULES 2.3 THE ADDITION RULES OF PROBABILITY PROBABILITY – numerical measure of the likelihood that a specific event will occur

CLASSICAL APPROACHES

EMPIRICAL SUBJECTIVE

CLASSICAL PROBABILITY P (E ) =

n( E ) n(S )

n(E) = number of successful outcomes n(S) = number of possible outcomes.

EXAMPLE: Find the probability of obtaining an even number in one roll of a die.

SOLUTION: Let E = obtain even numbers

P(E)=

Number of outcomes include in E 3 = = 0.5 Total number of outcomes 6

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CHAPTER 2: PROBABILITY & COUNTING RULES EMPIRICAL PROBABILITY -

Relies on actual experiment to determine the likelihood of outcomes (based on observation).

-

If an experiment is repeated n times and an event E is observed f times, then according to the relative frequency concept of probability:

P( E ) =

frequency for the class f = total frequencies in the distribution n

Example: Type of blood: Type

Frequency

A

22

B

5

AB

2

O

21

Total

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Find the probability that person a)

has type B blood

b)

has type AB or A blood

c)

has neither type B nor type A blood

d)

does not have type O blood

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CHAPTER 2: PROBABILITY & COUNTING RULES SUBJECTIVE PROBABILITY Uses a probability value based on an educated guess or estimate, employing opinions and inexact information Example: weather forecasting, predicting outcomes of sporting events RULES OF PROBABILTY i.

0 ≤ P (A) ≤ 1 .

ii.

If A is cannot occur, then P(A)=0.

Example: When a single die is rolled, find the probability of getting an 8. iii.

If an event A is certain to occur, then P(A)=1.

Example: When a single die is rolled, find the probability of getting a number less than 9. iv.

The sum of the probabilities of the outcomes in the sample space is 1.

P (S ) = 1

∑

Example: Outcome (x) Probability, P(x)

1

2

3

4

5

6

1/6

1/6

1/6

1/6

1/6

1/6

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CHAPTER 2: PROBABILITY & COUNTING RULES UNIONS Given A and B are two events is a sample space S. The union of the two events A and B, denoted by the symbol A ∪ B, is the event containing all the elements that belong to A or B or both

INTERSECTION The intersection of two events A and B, denoted by the symbol A ∩ B, is the event containing all elements that are common to A and B

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CHAPTER 2: PROBABILITY & COUNTING RULES ADDITIVE RULES 

If A and B are two events, then

P( A ∪ B) = P( A) + P( B) − P( A ∩ B)

 If A and B are NOT MUTUALLY EXCLUSIVE, then

P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Note: P(A ∩ B) ≠ 0  If A and B are MUTUALLY EXCLUSIVE, then Note:

P(A ∪ B) = P(A) + P(B)

P(A ∩ B) = 0

MUTUALLY EXCLUSIVE : A and B cannot occur at the same time Mutual exclusive:

Not mutual exclusive:

Randomly selecting person with type A blood

Randomly selecting a female student

Randomly selecting person with type O blood

Randomly selecting a student who is a junior

Person cannot have type A and O blood at

Student can be both female and a junior

same time 

For three events A, B and C

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)

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CHAPTER 2: PROBABILITY & COUNTING RULES EXAMPLE: A city has 9 coffee shops: 3 Starbucks, 2 Caribou Coffees and 4 Crazy Mocha Coffees. If a person selects one shop at random to buy a cup of coffee, find the probability that it is either a Starbucks or Crazy Mocha Coffees.

SOLUTION: Since there are 3 Starbucks and 4 Crazy Mocha, and a total of 9 coffee shops, P (Starbucks or Crazy Mocha) = P (Starbucks) +P (Crazy Mocha) = The events are mutually exclusive.

3 4 7 + = 9 9 9

EXAMPLE: In a hospital unit, there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male.

SOLUTION Staff Females Males Total Nurses 7 1 8 Physicians 3 2 5 Total 10 3 13

P (Nurse or Male) = P (Nurse) + P (Male) – P (Male Nurse) =

The events are not mutually exclusive.

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8 3 1 10 + − = 13 13 13 13

CHAPTER 2: PROBABILITY & COUNTING RULES COMPLEMENTARY EVENTS The complementary of an event A is the set of outcomes in the sample space that are not included in the outcomes of event A. complement of A is denoted by the symbol A’ P(A’)=1 – P(A)

S

A A’

Example: If the probability that a student owns computer is 0.4, find the probability that a student does not own computers.

A ball is chosen at random from a box contain 7 balls, which two of the balls are green in color. What is the probability of getting a ball that not green in color?

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CHAPTER 2: PROBABILITY & COUNTING RULES 2.4 THE MULTIPLICATIVE RULES AND CONDITIONAL PROBABILITY The probability of an event B occurring when it is known that some event A has occurred is called a conditional probability and is denoted by P(B|A). The symbol P(B|A) is read as “the probability that B occurs given that A occurs” or simply “the probability of B, given A”

P( B | A) =

P( B ∩ A) P( A)

Example: Given that A and B are two events with probabilities P(A)=1/4, P(B)=1/3 and P(A|B)=1/2.

Find a)

P( A ∩ B)

b)

P( A ∪ B)

c)

P( B | A)

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CHAPTER 2: PROBABILITY & COUNTING RULES Example: The probability that a regularly scheduled flight departs on time is P (D) =0.83; the probability it arrives on time is P (A) =0.82; and the probability that it departs and arrives on time is P (D ∊ A) =0.78. Find the probability that a plane a) Arrive on time given that it departed on time

b) Departed on time, given that it has arrived on time

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CHAPTER 2: PROBABILITY & COUNTING RULES MULTIPLICATIVE RULES 

Two events A and B are two independent events if the event A has no influence on

the event B to occur.

Example: i) having a large shoe size and having a high IQ ii) tossing a coin and drawing a card from a desk 

Two events A and B are two dependent events if the event A has influence on the

event B to occur.

Example: i) having high grades and getting a scholarship ii) smoking excessively and having a lung cancer

If A and B are independent events, the probability of occurring is

P( A ∩ B) = P( A) P( B)

If A and B are dependent events, the probability of both occurring is

P( A ∩ B) = P( A | B) P( B) or = P( B | A) P( A)

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CHAPTER 2: PROBABILITY & COUNTING RULES Example: Two events A and B are such that P(A)=1/4, P(A|B)=1/2 and P(B|A)=2/3. a) Are A and B independent events? Why?

b) Are A and B mutually exclusive events? Why?

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CHAPTER 2: PROBABILITY & COUNTING RULES USEFUL NOTE

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CHAPTER 2: PROBABILITY & COUNTING RULES 2.5 BAYES’ THEOREM TREE DIAGRAM ~ Used to display the outcomes of an experiment which consist of a series of activities. ~ Ensure that all logical possibilities are considered.

Example: 1. Three companies, A, B and C are competing for a contract to build a condominium.

The probabilities that the companies A, B and C will win the contract are 0.25, 0.45 and 0.3 respectively. If companies A, B and C win the contract, the probabilities that they will make profits are 0.8, 0.9 and 0.7 respectively. Construct the tree diagram.

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CHAPTER 2: PROBABILITY & COUNTING RULES 2. The probability that Kasim, Ismail or Abu will be selected to represent Bank A at a particular negotiation is 0.35, 0.4 and 0.25 respectively. The probability that an agreement would be signed given that Kasim, Ismail or Abu is selected is 0.2, 0.5 and 0.3 respectively. Draw a tree diagram to show the above probability.

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CHAPTER 2: PROBABILITY & COUNTING RULES BAYES THEOREM

P( Br | A) =

P( B r ∩ A) k

=

P( B r ).P( A | B r ) k

∑ P( B ∩ A) ∑ P( B ) P( A | B ) i =1

i

i =1

i

i

Refer example 2: Given that an agreement would be signed, what is the probability that Kasim are selected to represent Bank A?

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