Question

You play a game with two six-sided dice. If you roll a sum of 3 or 8, you win ₱600. If you roll a sum of 10, you win 

₱400. However, you lose ₱500 for anything else. If you continue to play the game, how much do you expect to 

win or lose in the game?

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c:c} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 1+1 & 1+2 & 1+3 & 1+4 & 1+5 & 1+6 \\ \hdashline 2 & 2+1 & 2+2 & 2+3 & 2+4 & 2+5 & 2+6 \\ \hdashline 3 & 3+1 & 3+2 & 3+3 & 3+4 & 3+5 & 3+6\\ \hdashline 4 & 4+1 & 4+2 & 4+3 & 4+4 & 4+5 & 4+6 \\ \hdashline 5 & 5+1 & 5+2 & 5+3 & 5+4 & 5+5 & 5+6\\ \hdashline 6 & 6+1 & 6+2 & 6+3 & 6+4 & 6+5 & 6+6 \\ \hdashline \end{array}$$

There are $6^2=36$ pairs

$$1+2=2+1=3$$

$$2+6=3+5=4+4=5+3=6+2=8$$

$$4+6=5+5=6+4=10$$

$$E(X)=600(7/36)+400(3/36)-500(26/36)$$

$$=-7600/36\approx-211$$

I expect to lose ₱211 in the game.