Crossposted on Quant SE
Recently, I was asked the following question in an interview with a prop trading firm.
You are given the opportunity to make money by betting a total of 100 bucks on the outcome of two simultaneous matches:
- Match $A$ is between the Pink team and the Maroon team
- Match $B$ is between the Brown team and the Cyan team
The Pink team's probability of victory is $40\%$. The Brown team's probability of victory is $70\%$. The betting odds are
- Pink: 7:4
- Maroon: 2:3
- Brown: 1:4
- Cyan: 3:1
How much money do you bet on each team?
You do not have to bet all 100 bucks, but your bets must be whole numbers and the total of all five blanks (bets on the four teams and the unbet amount) must sum to 100. There is no single "correct" answer, but there are many "wrong" answers. As a reminder, a hypothetical team having 2:7 odds means that if you bet 7 on that team and they win, you get your 7 bucks bet back and win an additional 2 bucks.
I am struck between an arbitrage approach vs a Kelly Criterion or EV maximization approach. Also, I know that Match A has a clear Arbitrage opportunity (Implied odds are < 1) but not sure how to approach match B as there isn't an arbitrage here (Implied odds are > 1).
Should we chose to not bet at all in Match B? Or should we bet on Cyan since it has a higher EV than Brown and use Kelly Criterion to determine how much to bet on Cyan?
Also, we wouldn't just want to bet all in (\$100) on 1 team as there is a (1-p) chance of going all bust and making nothing even if the EV is > 0.