# A and B throw pair of dice alternately.A wins the game if he gets a total of 6 and B wins if she gets a total of 7.It A starts the game,find the probability of winning the game by A in third throw of the pair of dice.

Toolbox:
• A wins if he gets a total of 6 in a through of pair of dice
• Total of 6 can be obtained
• {$$(1\;5)(5\;1)(2\;4)(4\;2)(3\;3)$$}
• Total number of wins for throw of pair of dice=36
• B wins if he gets a total of 7
• Total of 7 can be obtainedby
• {$$(1\;6)(6\;1)(2\;5)(5\;2)(3\;4)(4\;3)$$}
• P(A winning)=1-P(A non winning or losing)
• P(B winning)=1-P(B non winning or losing)
A( wins in third throw)
=P(A loses in 1$$^{st}$$and B loses in 2$$^{ed}$$ and A wins in 3$$^{rd}$$)
=P(A winning)=$$\large\frac{5}{36}$$
=P(A loosing)=1-$$\large\frac{5}{36}\;=\;\frac{31}{36}$$
=P(B wins)$$\large\frac{6}{36}$$
=P(B loosing) =1-$$\large\frac{6}{36}$$=$$\large\frac{30}{36}$$
P(A wins in 3$$^rd$$ throw)
=$$\large\frac{31}{36}\;\times\;\frac{30}{36}\;\times\;\frac{5}{36}$$
=$$\large\frac{775}{7776}$$

edited Jun 4, 2013