Football teams `T_1 and T_2` have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of `T_1` winning. Drawing and losing a game against `T_2` are `(1)/(2),(1)/(6) and (1)/(3)` respectively. Each team gets 3 points for a win. 1 point for a draw and 10 pont for a loss in a game.
Let `X and Y` denote the total points scored by teams `T_1 and T_2` respectively. after two games.

Options

A. `(1)/(4)`
B. `(5)/(12)`
C. `(1)/(2)`
D. `(7)/(12)`

Answer

B

Explanation

Consider the following events :
`W_(ij)`=Team `T_(i)` wins in `j^(th)` game
`D_(i)=i^(th)` game is draw, i=1,2
`P(X gt Y)=P{(W_(11) cap W_(12)) cup (W_(11) cap D_(2)) cup (D_(1) cap W_(12))}`
`=P(W_(11) cap W_(12))+P(W_(11) cap D_(2))+P(D_(1) cap W_(12))`
`=P(W_(11))P(W_(12))+P(W_(11))P(D_(2))+P(D_(1))P(W_(12))`
`=(1)/(2)xx(1)/(2)+(1)/(2)xx(1)/(6)+(1)/(6)xx(1)/(2)=(5)/(12)`