if a gambler rolls two dice and gets a sum of 10,he wins 10$,and if he gets a sum of three, he wins 20$. The cost to play the game is 5$, what is the expectations of this game?

There are 36 = 6*6 total possible outcomes if two dice are rolled. For three out of them ({4, 6}, {5,5}, {6,4}) the sum is 10 and for two of them ({1,2} and {2,1}) the sum is 3.

The probabilities are P( sum = 10) = 3/36 = 1/12 and P(sum=3) = 2/36 = 1/18.

The expected value of winnings is 10* 1/12 + 20 * 1/18 = 5/6 + 10/9 = (15+20)/18 = 35/18 = $1.94, however, since a gambler has to pay $5 for the game, the total expectation is the loss of $3.06.