In each of the following questions answer the following What

In each of the following questions answer the following: What kind of random variable is this question asking about? Write a probability statement i.e. P(sr > X describing the question. What are the relevant pieces of information for the probability statement? What are the relevant pieces of information for the probability distribution function? Write down the formula for your solution in terms of the probability distribution function. Give your numerical answer and the relevant MATLAB code used to find the answer. This can be attached in a separate file as supplementary information. 1. Two teams, A and B. play a series of games. If team A has a probability 0.4 of winning each game. is it to their advantage to play the best of three out of five games or the best four out of seven, and why? Assume the outcomes of successive games are independent. 2. A multiple choice test consists of 20 items, each with four choices. A student is able to eliminate one of the choices on each question as incorrect and chooses randomly from the remaining three choices. A passing grade is 12 items or more correct. (a) What is the probability the student passes? (b) Answer the question in part (a) again assuming that the student can eliminate two of the choices on each question. 3. Suppose that a rare disease has an incidence of 1 in 1.000. Assuming that members of a population are affected independently, find the probability of k cases in a populations for k = 0.1.2. 4. Phone calls are received at your residence at a rate of two calls per hour. If you leave the residence for 30 minutes what is the probability you will miss a phone call? 5. If X N(Mu, sigma), show that P(|X - Mu|

Solution

1)x no of wins by A is binomial with p = 0.4

P(X=3 in 5) = 0.2304

P(x=4 in 7) = 0.1935

Hence 3 out of 5 is better.

2) NO of correct questions X is binomial with p = 1/3 and n = 20

a) Prob that student passes = P(X>=12)

= 0.01286

b) If he can eliminate two choices p changes to 0.5

Hence P(X>=12) = 0.2517

3) p = very small = 0.001

Hence x is Poisson

P(x=0) = 0.9990

P(x=1) = 0.00099

P(X=2) = 4.995x10^-7

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4) 2 calls per hour

(POizzon with lamda = 1 for 30 minuttes)

P(X=0) in 30 minutes = 0.3679

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5) Here z score becomes

|z|<=0.675

Hence prob = 0.2500(2)

= 0.5000