A gambler pays $5 to play a game of dice. he rolls two dice. if he rolls a sum of 10 the house pays him $10; if he rolls a sum of 3 the house pays him $20, otherwise he loses. what is the expectation of this game?

Rolling 2 dice there are 36 possible outcomes.

Sum of 10: 4, 6; 5, 5; 6, 4

3 outcomes have a sum of 10.

Sum of 3: 1, 2; 2, 1

2 outcomes have a sum of 3.

Each play costs $5.

If he rolls a sum of 10, he wins $10, but he paid $5 to play, so for each roll of a sum of 10 he earns $5.

If he rolls a sum of 3, he wins $20, but he paid $5 to play, so for each roll of a sum of 3 he earns $15.

For all the other 31 possible sums, he loses $5.

Add: 3 * $5 + 2 * $15 - 31 * $5 = $110

Now divide the sum by the total number of outcomes.

$110/36 = - $3.05555 ...

Answer: - $3.06