Making Sense of a Betting Market with Probabilities 🌱

Consider a game you play with your friend where you get $10ifyouwinandyoupayyourfriend$5 if you lose. How can you decide if you should play this game?

• Making a market comes down to expected value. In order for this to be a fair market, the expected value of playing this game (for either player) must be 0, otherwise the option with the lower expected value would not be picked.
• In the example, your expected value of playing the game $E\left(X\right)=10p-5(1-p)$ where $p$ is the probability that you win. For this game to be fair, $E\left(X\right)=0=10p-5(1-p)$ which implies that $p=1/3$.
• In fact, for a Bernoulli random variable like in this game, the probability for a fair market generalizes to $p=\left(\text{money from losing}\right)/\text{(money from winning + money from losing)}$. In the Bernoulli case, you should play the game if $p>\left(\text{money from losing}\right)/\text{(money from winning + money from losing)}$ and not play if it is lower. In this example, you should play the game if you think that $p<1/3$.
• You can also derive the fair betting probabilities for other distributions.
• The variance of this distribution $Var\left(X\right)=E\left(X^2\right)-E(X{)}^2=100p+25(1-p)-(10p-5(1-p))=70p+20$.