2. In a tennis match, two players, Alfred and Bud, play three sets, and the firs

Question

2. In a tennis match, two players, Alfred and Bud, play three sets, and the first player who wins 2 sets wins the match. Suppose Alfred has a 60% of chance of winning each set. If Alfred wins a set, we denote that outcome by A. So if Alfred wins first, loses 2nd and wins 3rd, the outcome will be {ABA}, etc.

(a) Fill in the sample space: S = {AA, BB, _______, ______, _______, _______ }(six outcomes total) and compute the corresponding probabilities of each simple outcome in S.

(b) Find the outcomes favorable to the event E = {Alfred wins the match} = { _____, _______, _____ }, and compute the probability of the event E. P(E)=

(c) Let X denote the number of sets the match has ( 2 or 3 ); Complete the distribution table for X.

X=x P(X=x )

2

3

P(X=2)=

P(X=3)=

(d) Find the population mean, µX , of X.

(e) Find the population variance, X 2 , of X, and the corresponding population standard deviation.

Explanation / Answer

Solution-

1. Sample space includes- AA, BB, ABA, BAA, BAB, ABB - 6 outcomes.

2. E- When alfred wins the match. Favourable outcomes- AA, ABA and BAA.

P(E) = 0.6*0.6 + 0.6*0.4*0.6 + 0.4*0.6*0.6 = 0.648

3. P(X=2) = 0.4 * 0.4 + 0.6 * 0.6 = 0.52

P(X=3) = 0.4*0.6*0.6+ 0.4 *0.6*0.4 + 0.6 *0.4*0.4 + 0.6 *0.4 *0.6 = 0.48

4. E(X) = 2*0.52 + 3*0.48

= 2.48

5. var(X) =E(X2) -[ E(X)]2

= 22*0.52 + 32*0.48 - 2.482

= 0.2496

and standard deviation = 0.24960.5

= 0.5

Answers

TY!

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