Probability And Counting Rules The relevant R codes and outputs Must Be
Probability and Counting Rules The relevant R codes and outputs must be attached for full credit. 1. Explain whether or not the following numbers could be examples of a probability. (8 points total 2 points each) a) P (A) = 0.5 b) P ( B ) = 0 c) P ( C ) = 1.6 d) P ( D ) = -3 2. A quality control engineer randomly selects two light bulbs from a shipment to determine if they meet all the specifications defined by a company. Let M = a light bulb meets all the specifications defined by a company, and N = a light bulb does NOT meet all the specifications defined by a company.
Write the sample space of this experiment in set notation. (1 point) 3. Consider x be an event defined by the waiting time (in hours) between successive speeders spotted by a radar unit. Write the sample space of this event using mathematical notation . (1 point) 4. Consider the sample space S = {copper, sodium, nitrogen, potassium, uranium, oxygen, zinc} and the events A = {copper} B = {sodium, nitrogen, potassium} C = {oxygen} List the elements of the sets corresponding to the following events using set notation : Source: Walpole, R.E., Myers, R.H., Myers, S.L., & Ye, K. (2012). Probability & Statistics for Engineers & Scientists .
Boston, MA: Pearson. a. BC (1 point) b. A B C (1 point) c. (A B)C (AC C) (1 point) d. B C (1 point) 5. A drug for the relief of asthma can be purchased from 5 different manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength.
How many different ways can a doctor prescribe the drug for a patient suffering from asthma? (1 point) 6. A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 different heating systems, a garage or carport, and a patio or screened porch. How many different plans are available to this buyer? (1 point) 7. How many distinct ways are there to arrange the letters of the word “Miami”? (Note: Assume capital M and lowercase m are the same letter) (1 point) Note: Students can use RStudio or a formula. 8.
A statistics class for engineers consists of 53 students. The students in the class are classified based on their college major and sex as shown in the following contingency table: College Major Sex Industrial Engineering Mechanical Engineering Electrical Engineering Civil Engineering Total Male Female Total If a student is selected at random from the class by the instructor to answer a question, find the following probabilities. Report your answer to 4 decimal places. Consider the following events: A: The selected student is a male. B: The selected student is industrial engineering major.
C: The selected student is electrical engineering major. D: The selected student is civil engineering major. Note: Indicate the type of probability as marginal, joint or conditional when asked. a) Find the probability that the randomly selected student is a male. Indicate the type of probability. (1 + 1 = 2 points) b) Find the probability that the randomly selected student is industrial engineering major. Indicate the type of probability. (1 + 1 = 2 points) c) Find the probability that the randomly selected student is male industrial engineering major.
Indicate the type of probability. (1 + 1 = 2 points) d) Given that the selected student is industrial engineering major, what is the probability that the student is male? Indicate the type of probability. (1 + 1 = 2 points) e) Based on your answers on part a and d, are sex and college major of students in this class independent? Provide a mathematical argument? (1 point) f) Consider the events A and B. Are sex and college major mutually exclusive events? Provide a mathematical argument to justify your answer. (1 point) g) Find the probability that the randomly selected student is male or industrial engineering college major. (1 point) h) Consider the events C and D.
Are college major mutually exclusive events? Provide a mathematical argument to justify your answer. (1 point) i) Find the probability that the randomly selected student is electrical or civil engineering college major. (1 point) j) What is the probability that a randomly selected student is neither a male nor an industrial engineering college major (1 point) 9. Police plan to enforce speed limits by using radar traps at four different locations within the city limits. The radar traps at each of the locations (location 1, location 2, location 3, and location 4, respectively) will be operated 40%, 30%, 20%, and 30% of the time. A person who is speeding on his/her way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations (location 1, location 2, location 3, and location 4, respectively). a) If a speeding person is randomly selected, what is the probability that he/she will receive a speeding ticket?
Report your answer to 4 decimal places. (5 points) Hint: Defining events and making a tree diagram would be very helpful. Students may use the following template to produce a tree diagram. Tree Diagram b) If this person received a speeding ticket on his/her way to work, what is the probability that he/she passed through the radar trap located at location 2? (2 points) 10. Assume a small town has two fire engines operating independently. The probability that a specific engine is available when needed is 0.94. a) Find the probability that both fine engines are available when needed. Report your answer to four decimal places. (1 point) b) What is the probability that neither fire engine is available when needed? Report your answer to four decimal places. (1 point) c) What is the probability that at least one fire engine is available
when needed)? Report your answer to four decimal places. (2 points) 11. The probability that a doctor correctly diagnoses a particular illness is 0.75. Given that the doctor makes an incorrect diagnosis, the probability that the patient files a lawsuit is 0.85.
What is the probability that the doctor makes an incorrect diagnosis and the patient sues? Report your answer to four decimal places. (2 points) 12. The probability that a patient survives from a delicate heart operation is 0.8. a) What is the probability that exactly 2 of the next 3 patients who have this operation survive? Report your answer to four decimal places. (2 points) b) What is the probability that all of the next 3 patients who have this operation survive? Report your answer to four decimal places. (2 points) c) What is the probability that all of the next 3 patients who have this operation die? Report your answer to four decimal places. (2 points)
Paper For Above instruction
In this comprehensive analysis, we explore fundamental concepts of probability, including the validity of probability values, sample space construction, set operations, combinatorial calculations, and probability application in real-world scenarios. We will use R programming language to illustrate calculations and results, ensuring clarity and accuracy in statistical reasoning.
Evaluating Probability Values
Probability values must adhere to the fundamental rule that probabilities are numbers between 0 and 1 inclusive. Therefore, only values within this range are valid probabilities. The given examples illustrate this:
P(A) = 0.5 is valid as it lies within [0,1].
P(B) = 0 is valid; it indicates an impossible event.
P(C) = 1.6 is invalid as it exceeds 1.
P(D) = -3 is invalid as it is negative.
In R, we could verify these with simple conditional checks or directly interpret the values. For example:
prob_A <- 0.5
prob_B <- 0
prob_C <- 1.6
prob_D <- -3
valid_prob_A <- (prob_A >= 0 & prob_A <= 1)
valid_prob_B <- (prob_B >= 0 & prob_B <= 1)
valid_prob_C <- (prob_C >= 0 & prob_C <= 1)
valid_prob_D <- (prob_D >= 0 & prob_D <= 1)
valid_prob_A
valid_prob_B
valid_prob_C
valid_prob_D
This confirms A and B are valid probabilities, whereas C and D are invalid.
Sample Space Construction
In the experiment with two light bulbs, where M indicates a bulb meets specifications and N indicates it does not, the sample space is the set of all ordered pairs:
S <- list(c("M", "M"), c("M", "N"), c("N", "M"), c("N", "N"))
This defines the four possible outcomes, assuming independence in selection.
For the radar speeders, the sample space can be represented as the set of possible waiting times, typically described by a continuous interval, such as:
space_x <- c(0, +Inf)
which indicates waiting times from zero to infinity hours.
Set Operations and Event Elements
Given the universal set S and subsets A, B, and C, the set operations in R can be applied using functions like intersect() and union(). For example:
S <- c("copper", "sodium", "nitrogen", "potassium", "uranium", "oxygen", "zinc")
A <- c("copper")
B <- c("sodium", "nitrogen", "potassium")
C <- c("oxygen")
# BC
BC <- setdiff(B, C)
# A B C (union of A, B, C)
union_ABC <- union(A, union(B, C))
# (A B)C (intersection of A and B, then set difference with C)
A_B <- intersect(A, B)
A_B_C <- setdiff(A_B, C)
# B C (intersection of B and C)
B_C <- intersect(B, C)
This demonstrates how to construct relevant event sets explicitly.
Combinatorial Calculations
To find the number of ways to prescribe drugs, consider the possible choices across categories:
5 manufacturers
3 forms (liquid, tablet, capsule)
2 strengths (regular, extra)
Number of prescriptions: 5 × 3 × 2 = 30.
Similarly, for the housing plans with design options, heating systems, garage/carport, and patio/screened porch:
Designs: 4
Heating: 3
Garage or carport: 2 options
Patio or screened porch: 2 options
Number of plans: 4 × 3 × 2 × 2 = 48.
Permutations of Letters
The word "Miami" (with M and m considered the same) has a total of 5 letters. Since M and m are identical, we consider the counts: letters <- c("M", "i", "a", "m", "i")
# Count frequency of each letter
table(letters)
# Total arrangements considering identical letters
factorial(5) / (factorial(2) * factorial(2))
This yields the total distinct arrangements.
Probability in the Context of Engineering and Medical Data
Analyzing the statistics class, probabilities are computed based on class size, such as: total_students <- 53
# Example: probability that a student is male
prob_male <- sum(MaleCount) / total_students
# For joint probability, e.g., male and industrial engineering
prob_male_industrial <- count_male_industrial / total_students
These calculations are further categorized into marginal, joint, and conditional probabilities based on the known counts and subset overlaps. For example, the probability a student is a male (A) is marginal; the probability a student is a male and an industrial engineering major is joint; and the probability a student is male given that they are in industrial engineering is conditional.
Application of Probabilities in Traffic and Fire Safety
Using tree diagrams or event-based calculations, the likelihood of receiving tickets or the availability of fire engines can be assessed. For example, the probability a speeding person gets a ticket encompasses the probability of passing through each location and getting caught, calculated via: probability_ticket <- sum(
location1_prob * pass_prob_location1,
location2_prob * pass_prob_location2,
location3_prob * pass_prob_location3,
location4_prob * pass_prob_location4
)
Conditional probabilities can be calculated when conditioned on the event of receiving a ticket, such as passing through location 2 given a ticket was received.
Medical Diagnosis and Survival Probabilities
Calculating combined probabilities involves multiplying the probability of each independent event. For example, the probability of a doctor making an incorrect diagnosis and the patient suing is: prob_incorrect_and_sues <- (1 - 0.75) * 0.85
Similarly, the chance that patients survive or die after surgery, modeled through binomial distribution, can be computed with R functions like dbinom().
Conclusion
This discussion has demonstrated the application of probability theory in various contexts from validating probability values, creating sample spaces, executing set operations, calculating combinatorial options, to analyzing real-world statistical data with R. These foundational principles enable rigorous decision-making and data interpretation in engineering, healthcare, urban planning, and public safety.
References
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Probability & Statistics for Engineers & Scientists
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