Table $3.6$shows a random sample of $200$cyclists and the routes they prefer. Let M= males and H= hilly path.

a. Out of the males, what is the probability that the cyclist prefers a hilly path?

b. Are the events "being male" and "preferring the hilly path" independent events?

The table shows a random sample of $200$cyclists and the routes they prefer. Let M= males and H= hilly path.

The total number of males $=90$

Out of total males, a number of males prefer the hilly path $=52$

So, the probability that the male cyclists who prefer a hilly path is calculated as

$$\begin{gathered} P\left(\text{male cyclists who prefers a hilly path}\right)=\frac{52}{90} \\ =0.58 \end{gathered}$$

Let, M = males

H = hilly path

To prove the independence of the events M and H, we must prove the following three conditions:

$$\begin{gathered} P(H\mid M)=P\left(M\right) \\ P(M\mid H)=P\left(H\right) \\ P(MANDH)=P\left(M\right)P\left(H\right) \end{gathered}$$

We have the probability that the cyclist prefers a hilly path is $0.58$.

So, $P(H\mid M)=0.58$

Also,$P\left(M\right)=0.45$

Since $P(H\mid M)\ne P\left(M\right)$

As a result, the first requirement of independency is not met.

As a result, the events are not self-contained.