# A and B tak

Given that A and B throws two dice.

The first who throw 9 awarded a prize.

The possibilities of getting 9 on throwing two dice are: P(S9) = P(getting sum 9)  P(SN) = P(not getting sum 9)  Let us assume A starts the game, A wins the game only when he gets 9 while throwing dice in 1st,3rd,5th,…… times

Here the probability of getting sum 9 on throwing a dice is same for both the players A and B

Since throwing a dice is an independent event, their probabilities multiply each other

P(Awins) = P(S9) + P(SN)P(SN)P(S9) + P(SN)P(SN)P(SN)P(SN)P(S9) + ……………  The series in the brackets resembles the Infinite geometric series.

We know that sum of a infinite geometric series with first term ‘a’ and common ratio ‘o’ is .    P(Bwins) = 1-P(Awins)   P(Awins):P(Bwins) = 9:8

Thus proved

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