Betting Game 1
In an enchanted math realm, a mystical mathematician offers you a game with a standard 6-sided die. You're given a chance to bet on the outcome being even or odd. Here's the catch: For every 4 dollars you bet on the outcome being even, if you're correct, you'll receive 6 dollars back. On the other hand, for every 6 you bet on the outcome being odd, a win gives you 9 dollars back. If you play this game optimally, how much should you bet?
Answer
Betting on Even:
\[
EV_{\text{even}} = \frac{1}{2}(2) + \frac{1}{2}(-4) = 1 - 2 = -\$1
\]
Betting on Odd:
\[
EV_{\text{odd}} = \frac{1}{2}(3) + \frac{1}{2}(-6) = 1.5 - 3 = -\$1.5
\]
Since both individual bets yield a negative expected value, any linear combination of them would also result in a negative expected profit. Hence the optimal strategy is to not bet at all.