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Question

In a hockey match, both teams A and B scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.

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Solution

Let A: team A is declared as a winner

Let B: team B is declared as a winner.

Let E: die shows six on the throw.

Clearly, P(E)=16

P(¯¯¯¯E)=1P(E)=116=56

If captain of team A starts then he may get a six in 1st throw or 3rd throw or 5th throw and so on.

Therefore, P(A)=P(E)+P((¯¯¯¯E)(¯¯¯¯E)(E))+P((¯¯¯¯E)(¯¯¯¯E)(¯¯¯¯E)(¯¯¯¯E)E)+......

Using sum of infinite G.P. Sinfinite=a1r

=16+(56)216+(56)416+.....=1612536

i.e., P(A)=611 and P(B)=1P(A)=1611=511

The decision of referee wasn't fair since team A has more chances of being declared a winner despite the fact that both the teams had secured same number of goals.


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