Exacta In a horse race, an exacta bet is won by correctly selecting the horses that finish first and second, and they must be in the order that they finished. The 140th Kentucky Derby race was run with 19 horses. If you make random selections for an exacta bet, what is the probability of winning?

To win a bet, two horses that finish first and second need to be selected out of 19 horses such that the order of the selection matters.

The permutation rule applies to a condition when the order of selecting the items from a batch is important.

The formula to select r items from n is shown below:

${}^n{P}_r=\frac{n!}{\left(n-r\right)!}$

Here, the total number of horses is 19.

The number of horses to be selected is 2.

Since the order of the selection of the horses is important, the permutation rule is used.

The total number of ways in which two horses can be selected is equal to:

$$\begin{gathered} {}^{19}{P}_2=\frac{19!}{\left(19-2\right)!} \\ =342 \end{gathered}$$

The number of ways of selecting the two horses that finish first and second in the race is 1.

Probability is the ratio of supportive outcomes to an event over the total number of outcomes feasible.

The probability of winning the bet is equal to:

$P\left(\text{winning}\text{the}\text{bet}\right)=\frac{1}{342}$

Therefore, the probability of winning the bet is equal to $\frac{1}{342}$.