Making Sense of a Betting Market with Probabilities 🌱

Consider a game you play with your friend where you get $10 if you win and you pay your friend ​$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 \(E(X) = 10p - 5(1-p)\) where \(p\) is the probability that you win. For this game to be fair, \(E(X) = 0 = 10p - 5(1-p)\) 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 \(p = (\text{money from losing})/\text{(money from winning + money from losing)}\). In the Bernoulli case, you should play the game if \(p > (\text{money from losing})/\text{(money from winning + money from losing)}\) 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 \(Var(X) = E(X^2)-E(X)^2 = 100p + 25(1-p) - (10p - 5(1-p)) = 70p + 20\).