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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.` <br> The probability of A's winning by the score of (3,1) os { (A loses I, wins II, II, IV sets) <br> +P(A wins I, loses II and wins III, IV sets) <br> +P(A sins sets I, II loses III and wins set IV) 2 <br> `=1/2xx1/4xx1/2xx1/2xx1/2((1)/(2))((1)/(4))((1)/(2))+1/2xx1/2xx1/2xx1/4=2/32` <br> The probability of A' s winning by the score of (3,2) is <br> `{:("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",):}` <br> `=1/3((3)/(4))((1)/(4))((1)/(2))^(2)+1/2((1)/(4))((1)/(2))((1)/(4))((1)/(2))` <br> `+1/2((1)/(4))((1)/(2))((1)/(2))((1)/(4))+1/2((1)/(2))((3)/(2))((1)/(4))((1)/(2))` <br> `+1/2((1)/(2))((1)/(4))((1)/(2))((1)/(4))+1/2((1)/(2))((1)/(2))((3)/(4))1/4` <br> `=3/128+1/128+1/128+3/128+1/128+3/128+12/128` <br> Therefore, the probability that A wins that match is <br> `1/8+3/32+12/128+(16+12+12)/(128)=40/128=5/16`**Introduction and definition**

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