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In a hockey match, both teams A and B scored the same number of goals up to the end of this game. 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, then the probability of B winning the match is:

Options:
a. 1​/6
b. 5/12​
c. 1/2​
d. 6/11​

1 Answer

3 votes

The probability of team B winning the match in this scenario is c) 1/2.

How can you solve the probability of B winning the match?

Both teams have scored the same number of goals, so the match is tied.

The winner is decided by a die roll, where the first captain to get a six wins.

Since Captain A throws first, there are two possibilities:

Team A wins on their first throw: This happens if Captain A throws a six on the first roll. The probability of this is 1/6.

Team B wins on one of their throws: If Captain A doesn't get a six on the first try, Captain B has the chance to win on any of their remaining throws. The probability of this happening depends on how many throws Captain A takes before getting a six.

Therefore, the total probability of B winning the match (considering all possible scenarios where A doesn't win on the first throw) is:

1/6 * (1 + 1/6 + 1/18 + ...)

This is an infinite geometric series with a first term of 1/6 and a common ratio of 1/6. The sum of this series is:

1/6 / (1 - 1/6) = 1/2

Therefore, the overall probability of B winning the match is:

Team B wins on first throw + Team B wins on any subsequent throw

= 1/6 + 1/2

= 3/6

= 1/2

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