$6$
You want to invent a gambling game in which a person pays \$1 to enter and rolls two dice. If the sum is $7$, the person earns amount $c$ but doesn't get the dollar back for total earnings $c-1$. If the sum is not $7$, he loses his dollar for total earnings $-1$. For this to be a fair game, with expected earnings of $0$, what should be the value of $c$?
$6$
Let us denote by $X$ the random variable that registers the amount a player wins. Then the support of $X={c-1,-1}$. For this game, $|\Omega|=36$, so $$P(X=c-1)=P(\mbox{the sum is } 7)=P({(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)})=\frac{6}{36}=\frac{1}{6}.$$
Also, $$P(X=-1)=P(\mbox{the sum is not } 7)=1-\frac{1}{6}=\frac{5}{6}.$$
Therefore, $$E[X]=(c-1)\times\frac{1}{6}+(-1)\times\frac{5}{6}=\frac{c-6}{6}.$$
In order for $E[X]$ to be $0$, $c$ has to be $6$.