Question

Five horses are in a race. Mr. $A$ selected two of the horses at random and bet on them. The probability that Mr. $A$selected the winning horse is

• A. $\frac{4}{5}$
• B. $\frac{3}{5}$
• C. $\frac{1}{5}$
• D. $\frac{2}{5}$

Answer 1

The correct option is D

$\frac{2}{5}$

Explanation for the correct answer:

Step 1: Find the number of possible outcomes for a given experiment.

We have been given five horses in a race. Mr. $A$ selected two of the horses at random and bet on them.

We need to find the probability that Mr. $A$ selected the winning horse.

Out of five horses, Mr. $A$ selects two of the horses at random.

$$\begin{gathered} \Rightarrow n\left(S\right)={}^{5}C_2 \\ \Rightarrow n\left(S\right)=\frac{5!}{2!(5-2)!} \\ \Rightarrow n\left(S\right)=10 \end{gathered}$$

Step 2: Find the number of possible outcomes for a given event.

Mr. $A$ selected two of the horses at random and bet on them.

Let event $M$: Mr. $A$ selected the winning horse

Suppose Mr. $A$ has been selected pair of horses to bet and out of these two horses, one horse is the winning horse.

Out of remaining $4$we can select horse as,

$$\begin{gathered} \Rightarrow n\left(A\right)={}^{4}C_1 \\ \Rightarrow n\left(A\right)=\frac{4!}{1!(4-1)!} \\ \Rightarrow n\left(A\right)=4 \end{gathered}$$

Step 3: Find the required probability.

the probability that Mr. $A$ selected the winning horse is,

$$\begin{gathered} \Rightarrow p\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)} \\ \Rightarrow p\left(A\right)=\frac{4}{10} \\ \Rightarrow \mathit{p}\mathbf{\left(}\mathit{A}\mathbf{\right)}\mathbf{=}\frac{\mathbf{2}}{\mathbf{5}} \end{gathered}$$

Therefore, option (D) is the correct answer.