In a hockey match, both teams A and B scored same number of goals up t

1.	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.
Answer» Let E₁, E₂ be two events such that E₁ = the captain of team ‘A’ gets a six. E₂ = the captain of team ‘B’ gets a six. Here P (E₁) = 1/6 , P (E₂) = 1/6 P(E₁) = 1 - 1/6 = 5/6, P(E₂) = 1 - 1/6 = 5/6 Now P (winning the match by team A) = 1/6 + 5/6 x 5/6 x 1/6 + 5/6 x 5/6 x 5/6 x 5/6 x 1/6 + .... = 1/6 + (5/6)² x 1/6 + (5/6)⁴ x 1/6 + ... = (1/6)/(1 - 25/36) = 6/11 ∴ P (winning the match by team B) = 1 - 6/11 = 5/11 The decision of re free was not fair because the probability of winning match is more for that team who start to throw dice.

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.

Answer»

Let E₁, E₂ be two events such that

E₁ = the captain of team ‘A’ gets a six.

E₂ = the captain of team ‘B’ gets a six.

Here P (E₁) = 1/6 , P (E₂) = 1/6

P(E₁) = 1 - 1/6 = 5/6, P(E₂) = 1 - 1/6 = 5/6

Now P (winning the match by team A)

= 1/6 + 5/6 x 5/6 x 1/6 + 5/6 x 5/6 x 5/6 x 5/6 x 1/6 + ....

= 1/6 + (5/6)² x 1/6 + (5/6)⁴ x 1/6 + ...

= (1/6)/(1 - 25/36) = 6/11

∴ P (winning the match by team B) = 1 - 6/11 = 5/11

The decision of re free was not fair because the probability of winning match is more for that team who start to throw dice.