The expectation of this game is that the house (casino) takes in roughly $3.83 every time someone plays, and after enough plays, they will typically always win.
We can determine this case by looking at all of the possibilities and how much you can win or lose off of each. There are 36 total cases for what can happen when we roll the dice. Of those 36 cases, 9 of them produce positive winnings and 27 of them produce losses.
To calculate the winnings, we need to look at what type they are. 6 of them will be 7's which earn the gambler $20. 3 of them would be 4's, which earns the gambler $40.
6($20) + 3($40)
$120 + $120
$240
Then we look at the losses. This is easier to calculate since every time the gambler loses, he losses exactly $14. There are 27 of these instances.
27($14)
$378
Now we can look at the average loss per game by subtracting the losses from the gains and finding the average.
(Winnings - losses)/options
($240 - 378)/36
$3.83