<s>[INST] Optimal wager on two games given probability and odds Question. Suppose I have two games I would like to wager \$100 dollars on, but that I'm not required to wager the full \$100, i.e. I can bet \$50 if I want.
The games: $A$ vs. $B$ and $C$ vs. $D$, where the probabilities of each team winning are
$$P(A) = \frac{7}{10},\quad P(B) = \frac{3}{10}\quad\quad\text{and}\quad\quad P(C)=\frac{1}{4},\quad P(D) = \frac{3}{4}.$$
I am quoted the following odds for each game:
$$
\begin{array}{|c|c|}
\hline
\text{team} & \text{odds} \\ \hline
A & 2:7 \\ \hline
B & 3:1 \\ \hline
C & 7:2 \\ \hline
D & 1:3 \\ \hline
\end{array}.
$$
For example, if we bet on team $A$ we would need to wager \$7 to win \$2.
How should you bet?

So far. It seems to me that there may be several "right" answers, but many "wrong" answers. Initially, my idea was to just look at the individual expected payoffs:
$$
\mathbb{E}A = \frac{7}{10}(2) + \frac{3}{10}(-7) = \frac{-7}{10}\\
\mathbb{E}B = \frac{3}{10}(3) + \frac{7}{10}(-1) = \frac{1}{5} \\
\mathbb{E}C = \frac{1}{4}(7) + \frac{3}{4}(-2) = \frac{1}{4} \\
\mathbb{E}D = \frac{3}{4}(1) + \frac{1}{4}(-3) = 0$$
So we see that we should primarily wager on $B$ and $C$. As far as how much to allocate, I guess we could just scale the expected values of $B$ and $C$ such that we get
$$
\frac{1/5}{1/5 + 1/4} = \frac{4}{9}, \quad\quad \frac{1/4}{1/5 + 1/4} = \frac{5}{9}.
$$
So we wager about \$44 on $B$ and \$56 on $C$. Is this line of thinking correct, or is there actually an optimal wager that isn't what I have?
 [/INST] If the player is infinately risk averse she should bet $x$ on $C$ and $100-x$
on $D$ where $x$ is determined by the equation
\begin{eqnarray*}
\frac{x}{2}\ast 7+x &=&\frac{100-x}{3}+100-x\rightarrow  \\
x &=&\frac{160}{7}
\end{eqnarray*}
Then, if $C$ wins she will get a net profit of $\frac{160}{7}\frac{7}{2}+%
\frac{160}{7}-100=2.8571.$
And if $D$ wins she will get a net profit of $\ \frac{100-\frac{160}{7}}{3}-(%
\frac{160}{7})=2.\,857\,1$
Hence, there is a possibility to make a guaranteed profit of $%
2.\,8571$ for every $100$ invested. It is thus a good strategy
if she had a credit card without limit.
</s>