SOLUTION: Roger and Andy play a tennis match in which the first person to win two sets wins the match. The probability that Roger wins the first set is 0.6. For sets after the first, the pro

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Question 1203262: Roger and Andy play a tennis match in which the first person to win two sets wins the match. The probability that Roger wins the first set is 0.6. For sets after the first, the probability that Roger wins the set is 0.7 if he won the previous set, and is 0.25 if he lost the previous set. No set is drawn.
(i) Find the probability that there is a winner of the match after exactly two sets.
(il) Find the probability that Andy wins the match given that there is a winner of the match after
exactly two sets.
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52858) About Me  (Show Source):
You can put this solution on YOUR website!
.
Roger and Andy play a tennis match in which the first person to win two sets wins the match.
The probability that Roger wins the first set is 0.6.
For sets after the first, the probability that Roger wins the set is 0.7 if he won the previous set,
and is 0.25 if he lost the previous set. No set is drawn.
(a) Find the probability that there is a winner of the match after exactly two sets.
(b) Find the probability that Andy wins the match given that there is a winner of the match after
exactly two sets.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

(a)  The winning record is either RR or AA.

     P(RR or AA) = P(RR) + P(AA).    ( <<<---===  the events RR and AA are disjoint )

     P(RR) = P(R wins 1st set)*P(R wins 2nd set) = 0.6*0.7 = 0.42.

     P(AA) = P(A wins 1st set)*P(A wins 2nd set) = P(R loses 1st set)*P(R loses 2nd set) = 

                                                 =     (1-0.6)*(1-0.25) = 0.4*0.75 = 0.3.   

     Therefore,  P(RR or AA) = 0.42 + 0.3 = 0.72.    ANSWER



(b)  Question (b) asks to find the probability of event (AA) given event (RR or AA).

     It is conditional probability, and from the calculations in part (a) the answer is OBVIOUS


        P( (AA) given (RR or AA) ) = P%28AA%29%2F%28P%28RR%29%2BP%28AA%29%29 = 0.3%2F0.72 = 30%2F72 = 5%2F12.   ANSWER

Solved.



Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!


(i) Find the probability that there is a winner of the match after exactly two

sets.



P(RR or AA) = P(RR or R'R') = P(RR) + P(R'R') = (0.6)(0.7) + (1-0.6)(1-0.25) =



0.42 +(0.4)(0.75) = 0.42 + 0.3



(il) Find the probability that Andy wins the match given that there is a winner of

the match after exactly two sets.



P[AA | (RR or AA)] = P[AA and (RR or AA)]/P(RR or AA) = P(AA)/0.72 = 0.3/0.72 =

30/72 = 5/12



Edwin