On a long train journey, a statistician is invited by a gambler to play a dice game. The game uses two ordinary dice which the statistician is to throw. If the total score is 12, the statistician is paid 6 by the gambler. If the total score is 8, the statistician is paid 3 by the gambler. However if both or either dice show a 1, the statistician pays the gambler 2. Let EX be the amount paid to the statistician by the gambler after the dice are thrown once. Determine, the probability that

(a) X-6,

(b) X 3,

(c) X--2 Find the expected value of X and show that, if the statistician played the game 100 times, his expected loss would be 2.78, to the nearest penny. Find the amount, fa, that the 6 would have to be changed to in order to make the game unbiased.