Question: Given 2 fair coins, I am going to toss them simultaneously. If both coins give heads, I lose $\$6$. if both coins give tails, I loss $\$4$. If they have different outcomes, say one is tail another one is head, then I gain $\$5$. If I toss the 2 coins many times, will I win or lose money?
My attempt:
Expected value is zero:
$$\frac{1}{4}\times (-6) + \frac{1}{4}\times (-4) + \frac{1}{2}\times 5 = 0.$$ Does this mean that I will neither win nor lose money, according to law of large numbers?
After checking the answer, it seems that I need to calculate variance as well as it will go to infinity as sample size increases.
Will the variance affect anything?