# How much money do you bet on each team?

From a trading interview at Optiver, I have a probability question that apparently has multiple correct answers. I would like to know if mine is correct.

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.

My solution

Equation of expected payoff

$$\left(P+ \frac{7}{4}P\right)0.4 + \left(M + \frac{2}{3}M\right)0.6+\left(B+\frac{1}{4}B\right)0.7+(C+3C)0.3 + R(unbet)$$

By looking at this, I concluded that betting on $$B$$ is not a good strategy so $$B = 0$$. Now the problem has become to maximize the following

$$1.1P + M + 1.2C + R$$

Since $$P+M+C+R = 100$$, the final problem has reduced to maximizing $$0.1 P + 0.2 C$$. Is my approach correct?

• How do you argument (mathematically) that $B$ is bad? Because for $B$ you get an average of $1.25\cdot 0.7=0.875$ of your input but for $M$ you get $1.666\cdot 0.6=1$ as a reward? Sep 1, 2022 at 7:15
• I am betting B amount on team B , If expected value is less than B why would I bet on that? That is why I am not betting on B Sep 1, 2022 at 7:19