In Exercises 5–36, express all probabilities as fractions.

Quinela In a horse race, a quinela bet is won if you selected the two horses that finish first and second, and they can be selected in any order. The 140th running of the Kentucky Derby had a field of 19 horses. What is the probability of winning a quinela bet if random horse selections are made?

Out of 19 horses, two will be selected in any order that finishes the first and second in the race.

The formula of combination is used to computethe number of ways in which r units can be selected out of n units irrespective of the order of draws.

The following is the formula:

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

Let A be the event of winning the bet.

The total number of horses is 19.

The selection is made in any order.

The total number of ways of selecting two horses out of the 19 horses is as follows:

$$\begin{gathered} {}^{19}{C}_2=\frac{19!}{\left(19-2\right)!2!} \\ =\frac{19\times 18}{2\times 1} \\ =171 \end{gathered}$$

The number of ways to select the two horses which finish the race at first and second position is 1( in any order).

The probability of winning the bet is computed below:

$$\begin{gathered} P\left(A\right)=\frac{\text{Number}\text{of}\text{favorable}\text{outcomes}\text{for}\text{A}}{\text{Total}\text{number}\text{of}\text{outcomes}} \\ =\frac{1}{171} \end{gathered}$$

Therefore, the probability of winning the bet is equal to $\frac{1}{171}$.