Q. 265.0( 4 Votes )

A, B and C in order toss a coin. The one to throw a head wins. What are their respective chances of winning assuming that the game may continue indefinitely?

Answer :

Given that A, B and C toss a coin until one of them gets a head to win the game.


Let us find the probability of getting the head.


P(AH) = P(A getting a head on tossing a coin)



P(AN) = P(A not getting head on tossing a coin)



P(BH) = P(B getting a head on tossing a coin)



P(BN) = P(B not getting head on tossing a coin)



P(CH) = P(C getting a head on tossing a coin)



P(CN) = P(C not getting head on tossing a coin)



It is told that A starts the game.


A tosses in1st,4th,7th,…… tosses.


This can be shown as follows:


P(WA) = P(A wins the game)


P(WA) = P(AH) + P(ANBNCNAH) + P(ANBNCNANBNCNAH) + …………………


Since tossing a coin by each person is an independent event, the probabilities multiply each other.


P(WA) = (P(AH)) + (P(AN)P(BN)P(CN)P(AH)) + ………………




The terms in the bracket resembles the infinite geometric series sequence:


We know that the sum of a Infinite geometric series with first term ‘a’ and common ratio ‘r’ is






B tosses in2nd,5th,8th,…… tosses.


This can be shown as follows:


P(WB) = P(B wins the game)


P(WB) = P(ANBH) + P(ANBNCNANBH) + P(ANBNCNANBNCNANBH) + …………………


Since tossing a coin by each person is an independent event, the probabilities multiply each other.


P(WB) = (P(AN)P(BH)) + (P(AN)P(BN)P(CN)P(AN)P(BH)) + ………………




The terms in the bracket resembles the infinite geometric series sequence:


We know that the sum of a Infinite geometric series with first term ‘a’ and common ratio ‘r’ is






P(WC) = P(winning of C)


P(WC) = 1-P(WA)-P(WB)




The chances of winning of A,B and C are .


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