I am new to the idea of expectation, and I want to make sure that I understand it. I came across this question stating:
You are playing a version of poker where you are dealt five cards and you would win $100 if you are dealt a three of a kind. Otherwise you will lose and you have to pay some amount of money. How much money should you have to pay to make this game a fair game (i.e. have expected value \$0)?
So if I have understood expectation correctly, the expectation of this kind of poker game would be: $$ \mathbb{E}[x] = -m\left(1-\frac{{13 \choose 1} \cdot {4 \choose 3} \cdot {12 \choose 2} \cdot {4 \choose1}^2}{52\choose 5}\right) +100\left(\frac{{13 \choose 1} \cdot {4 \choose 3} \cdot {12 \choose 2} \cdot {4 \choose1}^2}{52\choose 5}\right) $$ Where $m$ is the amount of money that we have to pay in this game, then I would set the expectation to zero and solve for $m$?
Yes, if $p$ is the probability of success you get $$ 0 = \mathbb{E}[X] = 100p-m(1-p) \iff m = \frac{100p}{1-p} $$ and it only remains to find $p$...