Question

if you are given odds of 2 to 3 in favor of winning a bet, what is the probability of winning the bet? the probability is (type an integer or a simplified fraction.)

Explanation:

Step1: Recall odds - probability relationship

The odds in favor of an event $E$ is defined as $\frac{P(E)}{1 - P(E)}$. Given odds of $a$ to $b$ in favor of winning, we have $\frac{P(E)}{1 - P(E)}=\frac{a}{b}$. Here $a = 2$ and $b = 3$.

Step2: Solve for $P(E)$

We start with $\frac{P(E)}{1 - P(E)}=\frac{2}{3}$. Cross - multiply to get $3P(E)=2(1 - P(E))$. Expand the right - hand side: $3P(E)=2 - 2P(E)$. Add $2P(E)$ to both sides: $3P(E)+2P(E)=2$, so $5P(E)=2$. Then $P(E)=\frac{2}{5}$.

Answer:

$\frac{2}{5}$