- Should I play a game that returns $1.5\times$ my wager if I win and returns nothing if I lose at exactly $50-50$ odds?
- How does this change if my odds are $>50\%?$
- How does this change if the payout is greater?
I'm trying to figure out which variable matters, if not both.
I've tried justifying my intuition (always play if $>50\%$ odds, regardless of payout) with references around, but most of the material around applies to gambling, where the payout is $2.0\times$.
I'd like a formula that accepts both of these variables as input and simply states "yes" or "no" ($1$ or $0$, any boolean, etc).
Suppose your wager is $x$ and, the probability that you win is $p$. Suppose the game returns your wager $k$ times if you win, and nothing if you lose (and I assume, that you pay your initial wager in both cases). Then your expected surplus will be:
$$(k-1)px - (1-p)x = (kp - 1)x$$
It is positive iff $p \geq \frac{1}{k}$.
Thus, in your particular case ($k = \frac{3}{2}$) the game will be worth playing if and only if your odds are greater than $\frac{2}{3}$