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)
It is given A starts the game, A wins the game only when he gets 6 while throwing die in 1st, 3rd, 5th,…… 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(Awins ) = P(S6) + P(SN)P(SN)P(S6) + P(SN)P(SN)P(SN)P(SN)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 s∞ = \(\cfrac{a}{1-r}\)
⇒ P(Bwins ) = 1 - P(Awins )
Since the probabilities if winnings of A and B are not equal, the decision of the referee is not fair