A tennis match of best of 5 sets is played by two players A and B. The probability that first set is won by A is 1/2 and if he losed the first, then probability of his winning the next set is 1/4, otherwise it remains same. Find the probability that A wins the match.

Explanation

In the tennis match of best of 5 sets, A can win the match, if score of A against the score of B is `(3,0),(3,1)or (3,2).` The probability of A's doing the score of (3,0) is `1/2xx1/2xx1/2=1/8.`
The probability of A's winning by the score of (3,1) os { (A loses I, wins II, II, IV sets)
+P(A wins I, loses II and wins III, IV sets)
+P(A sins sets I, II loses III and wins set IV) 2
`=1/2xx1/4xx1/2xx1/2xx1/2((1)/(2))((1)/(4))((1)/(2))+1/2xx1/2xx1/2xx1/4=2/32`
The probability of A' s winning by the score of (3,2) is
`{:("P(A losses I and II sets",),("+P(A losses I and III sets",),("+P(A losses I and IV sets",),("+P(A losses II and III sets",),("+P(A losses III and IV sets",):}`
`=1/3((3)/(4))((1)/(4))((1)/(2))^(2)+1/2((1)/(4))((1)/(2))((1)/(4))((1)/(2))`
`+1/2((1)/(4))((1)/(2))((1)/(2))((1)/(4))+1/2((1)/(2))((3)/(2))((1)/(4))((1)/(2))`
`+1/2((1)/(2))((1)/(4))((1)/(2))((1)/(4))+1/2((1)/(2))((1)/(2))((3)/(4))1/4`
`=3/128+1/128+1/128+3/128+1/128+3/128+12/128`
Therefore, the probability that A wins that match is
`1/8+3/32+12/128+(16+12+12)/(128)=40/128=5/16`