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 throw a die alternately and decide that the team, whose captain gets a first six, 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.

Question

Philoid · Accepted Answer

Given that teams A and B scored same number of goals.

It is asked captains of A and B to throw a die.

The first who throw 6 awarded a prize.

⇒ P(S₆) = P(getting 6)

⇒

⇒ P(S_N) = P(not getting 6)

⇒

It is given A starts the game, A wins the game only when he gets 6 while throwing die in 1^st,3^rd,5^th,…… times

Here the probability of getting 6 on throwing a die is same for both the players A and B

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

⇒ P(A_wins) = P(S₆) + P(S_N)P(S_N)P(S₆) + P(S_N)P(S_N)P(S_N)P(S_N)P(S₆) + ……………

⇒

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(B_wins) = 1-P(A_wins)

⇒

Since the probabilities if winnings of A and B are not equal, the decision of the referee is not fair