Answer only e) to h) problems a) to d) have the answer included

Problem 1: Relations among Useful Discrete Probability Distributions. A Bernoulli experiment consists of only one trial with two outcomes (success/failure) with probability of success p. The Bernoulli distribution

is

P (X = k) = p^kq^1-k, k=0,1

The sum of n independent Bernoulli trials forms a binomial experiment with parameters n and p. The binomial probability distribution provides a simple, easy-to-compute approximation with reasonable accuracy to hypergeometric distribution with parameters N, M and n when n/N is less than or equal to 0.10. In this case, we can approximate the hypergeometric probabilities by a binomial distribution with parameters n and p = M/N. Further, the Poisson distribution with mean μ = np gives an accurate approximation to binomial probabilities when n is large and p is small.

1. Suppose in a region in Saskatchewan, among a group of 20 adults with cancer, seven were physically abused during their childhood. A random sample of five adult persons is taken from this group. Assume that sampling occurs without replacement, and the random variable X represents the number of adults in the sample who were abused during their childhood period.

(a) Write the formula for p(x), the probability distribution of X. How this distribution is called?

P(X=x)=(5/x)(0.35)^x(1-0.35)^5-x

(b) Using the adequate formulas, find the mean and variance of X?

μ=1.75

variance = 1.1375

(c) Find the probabilities of all the possible values of X. Plot the histogram of X and try the locate the approximative value of the mean μ.

x 0 1 2 3 4 5

P(X=x): 0.116 0.3124 0.3364 0.1811 0.04877 0.005

(d) What is the probability that at least one person was abused during childhood?

0.884

Now suppose another survey in British Columbia reveals that among 180 adults with cancer, only 80 adults were abused in their childhood. Suppose again that a random sample of five adult persons is taken from this group without replacement and let denote by Y the random variable which represents the number of adults abused during their childhood period in the sample.

(e) Find the probabilities of all the possible values of Y and plot the histogram of Y. How do you compare this histogram with the histogram of X.

(f) Find the probabilities of all the possible values of Y using the formula for the binomial distribution with p = 80/180

as an approximation. Plot the histogram and compare it with the histogram obtained using the hypergeometric formula.

(g) Is the precedent approximation close enough? Why or why not?

(h) Calculate the mean and variance using both binomial and hypergeometric distributions, respectively. Provide a comparison and summarize your findings.

Jason has not been studying and believes that the probability that he will pass his chemistry exam is 0.4, the probability that he will pass the biology exam is 0.3 and the probability that he will pass both exams is 0.18.

1) Find the probability that Jason will pass at least one of the exams.

2)Find the probability that he will pass the chemistry exam given that he passes the biology exam.

3)Find the probability that he will pass the biology exam if he passes the chemistry exam.

4)Find the probability that he will fail both exams.

Please answer questions and explain answer!! Thank you

An electrical circuit contains two switches, labeled ―A‖ and ―B,‖ which are connected in parallel; i.e., the circuit functions if either (or both) of the two switches functions. The probability that switch A functions is 3/4. However, the switches do not operate independently. If switch A functions, then switch B functions with probability 8/9. If switch B functions, then switch A functions with probability 4/5. (a) What is the probability that both switches function? (b) What is the probability that switch B functions? (c) What is the probability that the electrical circuit functions? (d) Given that the electrical circuit functioned, what is the probability that switch B functioned?

Recent crime reports indicate that 17.3 motor vehicle thefts occur every hour in Canada. Assume that the distribution of thefts per hour can be approximated by a Poisson probability distribution.

a. Calculate the probability exactly four thefts occur in an hour.(Round the final answer to 5 decimal places.)

Probability            

b. What is the probability there are no thefts in an hour? (Round the final answer to 5 decimal places.)

Probability            

c. What is the probability there are at least 20 thefts in an hour? Use excel or online calculator to find the answer. (Round the final answer to 5 decimal places.)

Probability            

A diagnostic test has a 95% probability of giving a positive result when given to a person who has a certain disease. It has a 10% probability of giving a (false) positive result when given to a person who doesn’t have the disease. It is estimated that 15% of the population suffers from this disease.

(a) What is the probability that a test result is positive?

(b) A person recieves a positive test result. What is the probability that this person actually has the disease? (probability of a true positive)

(c) A person recieves a positive test result. What is the probability that this person doesn’t actually have the disease? (probability of a false negative)

Program #3

● Filename: pig.py

For this assignment, you’ll implement a dice game called PIG, pitting two human players against each other. PIG is played with a standard six-sided die and proceeds as follows:

• ● Each player starts with zero points, and the first to 20 points wins.

• ● The current player chooses to roll or hold.

• ● If they choose roll:

  • ○ The six-sided die is rolled

  • ○ The value of the die is added to their round points.

  • ○ The player goes again UNLESS...

  • ○ ...If the value on the die is 1, then the round is over and the player has lost all the points they

    accumulated in the round.

• ● If they choose hold:

  • ○ The points accumulated in the round are added to the player’s overall score.

  • ○ The round is over.

• ● When the round ends (either because the player rolled a 1 or they chose to hold), it becomes the other

  player’s turn unless somebody has won.

• ● We check to see if someone won at the end of each round (which is kind of unfair, because if Player

  One gets 20 points on the first round, then Player Two never gets a chance, but oh well tough for them).

  Requirements:

• ● Prompt each player to either R (roll) or H (hold).

• ● If they enter anything else, continue prompting them until they enter a correct input.

• ● On a roll, randomly generate a value 1-6 to represent the die.

• ● End the round when the die roll is value 1 (round points are lost), or the player chooses Hold (round

  points are added to that player’s overall score).

• ● End the game one either player has 20 or more points and announce the winner.

• ● Report everything as you go along -- what the die roll was, number of points so far, etc.

• ● As always with your CS5001 programs, your input/output must be friendly and informative.

  Helpful hints

  Python has a random module that can help you generate random numbers. You probably want to import random for this part of the assignment and call randint(...)

  import random # near the top of your file random.randint(1, 6) # or something like this in your program

  AMAZING points:

• ● Allow them to enter upper or lowercase letters. R/r for roll, and H/h for hold.

• ● Make your code extensible enough that we could add an arbitrary number of players with minimal

  coding-pain. (Consider using a list to store the player names and another list for the number of points each player has.)

A math class consists of 29 students, 17 female and 12 male. Three students are selected at random, one at a time, to participate in a probability experiment (selected in order without replacement).

(a) What is the probability that a male is selected, then two females?

   

(b) What is the probability that a female is selected, then two males?

   

(c) What is the probability that two females are selected, then one male?

   

(d) What is the probability that three males are selected?

   

(e) What is the probability that three females are selected?

   

An aeroplane company has submitted bids on two separate federal government defense contracts. The company president believes that there is a 44% probability of winning the first contract. If they win the first contract , the probability of winning the second is 65%. However, if they lose the first contract the president thinks that the probability of winning the second contract decreases to 53%.

a.What is the probability that they win both contracts?

b.What is the probability that they lose both contracts?

c.What is the probability that they win only one contract?

The probability of success is 0.68. A sample of 15 is taken. Assume independent trials. (Please show your work here)

1. What is the probability of succeeding all 15 times?

2. What is the probability of succeeding 9 times in 15 trials?

3. What is the probability of succeeding 14 times in 15 trials?

4. What is the probability of succeeding 12 or more times in 15 trials?

5. What is the probability of succeeding 11 or fewer times in 15 trials?