In a carnival game, you roll 2 dice. if the sum is 5, you receive a $6 payoff. if the sum is 10, you receive a $9 payoff. otherwise, you receive no payoff. what is the expected payoff? (round your answer to two decimal places.)

Expected value is the sum, for all possible outcomes, the value of each outcome times its probability.  In other words,
E[X]=sum  x*P(X=x)

For this given situation there are 4 outcomes (out of 36) that rolls a sum of 5 [1,4; 2,3; 3,2; 4,1], so p(5)=4/36=1/9 and its value is $6.  
There are 3 possible outcomes for a sum of 10 (again out of 36), so P(10)=3/36=1/12, with a value of $9.
All the other outcomes carry zero value, so the expected payoff is therefore
E[X]=sum x*P(x)
=$6*(1/9)+$9*(1/12)+$0(29/36)
=$(2/3+3/4)
=$17/12
=$1.42 (approx.)
So the expected payout is $1.42, not forgetting that it costs a certain amount to play the game.