Q. 285.0( 1 Vote )

A and B take turns in throwing two dice, the first to throw 10 being awarded the prize, show that if A has the first throw, their chance of winning are in the ratio 12:11.

Answer :

Given that A and B throws two dice.

The first who throw 10 awarded a prize.

The possibilities of getting 10 on throwing two dice are:

P(S10) = P(getting sum 10)

P(SN) = P(not getting sum 10)

Let us assume A starts the game, A wins the game only when he gets 10 while throwing dice in 1st,3rd,5th,…… times

Here the probability of getting sum 10 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(S10) + P(SN)P(SN)P(S10) + P(SN)P(SN)P(SN)P(SN)P(S10) + ……………

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) = 12:11

Thus proved

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