Mike and Jim will play tennis in \"best of 3\" sets format, but they will always

Question

Mike and Jim will play tennis in "best of 3" sets format, but they will always play the third set even though one of them has won the match by the end of the second set (unlike Problem 3 in Homework 2). Suppose that in each set, Mike and Jim may win with the same likelihood, independent with the results of previous sets. (It is like tossing a fair coin) What is the sample space Q in this case (use the same notations as Homework 2 Problem 3)? (b) Is it appropriate to apply a classical probability model on this sample space? Let A denote the event that Mike wins the match, and B denote the event that Jim wins the first set. What are the probabilities of the two events A and B Find the probability of the event C that Mike wins the first set but Jim wins the match.

Explanation / Answer

Sample space for this experiment is same as sample space of a coin tossing 3 times.

let,

M: Mike will win the set

and

J: Jim will win the set

a)

Sample space = {MMM,MMJ,MJM,MJJ,JMM,JMJ,JJM,JJJ}

b) We can use classical probabilty to get probability. Since we have total number of possible combinations is 8. If we consider any incident which occurs 2 times in sample space then probabilty of that incident is 2/8=1/4=0.25

b)

A: Mike wins the match.

B: Jim wins the first set.

P(A) = P(Mike wins the match)

P(A) = P(Mike win at least two sets)

P(A) = P(MMM OR MMJ OR MJM OR JMM)

P(A) = P(MMM)+ P(MMJ)+ P(MJM) + P(JMM)

P(A) = 1/8 + 1/8 + 1/8 + 1/8

P(A) = 4/8 =0.5

P(B) = P(Jim wins the first set)

P(B) = P(JMM OR JMJ OR JJM OR JJJ)

P(B) = P(JMM) + P(JMJ) + P(JJM) + P(JJJ)

P(B) = 1/8 + 1/8 + 1/8 + 1/8 = 4/8 = 0.5

c)

C: Mike wins the first set but Jim wins the match

P(C) = P(Mike wins the first set but Jim wins the match)

P(C) = P(Mikes wins first set and Jim wins reamaining two sets)

P(C)= P(MJJ)

P(C) = 1/8

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