Probability: Question 2

Worked solution

Part (a): Reading probabilities from the table

From the table, $60$ of the $120$ members attend Yoga, and $18$ of the $120$ members attend both Yoga and Spin: $P(\text{Yoga}) = \frac{60}{120} = 0.5 \qquad P(\text{Yoga and Spin}) = \frac{18}{120} = 0.15$

Part (b): Testing mutual exclusivity

Two events are mutually exclusive only if they can never happen together, i.e. if $P(A \cap B) = 0$.

Here, $P(\text{Yoga and Spin}) = 0.15 \neq 0$. Indeed, the table shows $18$ members do both. So the events are not mutually exclusive.

Part (c): Testing independence

Two events $A$ and $B$ are independent exactly when $P(A \cap B) = P(A) \times P(B)$.

First find $P(\text{Spin})$ from the table: $P(\text{Spin}) = \frac{42}{120} = 0.35$

Now compare the product with the actual joint probability: $P(\text{Yoga}) \times P(\text{Spin}) = 0.5 \times 0.35 = 0.175$ $P(\text{Yoga and Spin}) = 0.15$

Since $0.175 \neq 0.15$, the events are not independent (a member who attends Yoga is in fact slightly less likely to also attend Spin than a randomly chosen member).

Check by an alternative method: compare $P(\text{Spin} \mid \text{Yoga})$ with $P(\text{Spin})$. If the events were independent these would be equal. $P(\text{Spin} \mid \text{Yoga}) = \frac{18}{60} = 0.3 \qquad \text{but} \qquad P(\text{Spin}) = 0.35$ Since $0.3 \neq 0.35$, this confirms the events are not independent, agreeing with the product test above.

Part (d): Conditional probability

$P(\text{Spin} \mid \text{Yoga}) = \frac{P(\text{Yoga and Spin})}{P(\text{Yoga})} = \frac{0.15}{0.5} = 0.3$

This matches directly restricting attention to the $60$ Yoga members, of whom $18$ also attend Spin: $P(\text{Spin} \mid \text{Yoga}) = \frac{18}{60} = 0.3$

Consistency check: using the table probabilities, $P(\text{Yoga only}) = 0.5 - 0.15 = 0.35 = \frac{42}{120}$, $P(\text{Spin only}) = 0.35 - 0.15 = 0.2 = \frac{24}{120}$, and $P(\text{neither}) = 1 - 0.5 - 0.35 + 0.15 = 0.3 = \frac{36}{120}$. All of these match the table exactly, confirming every value used above is consistent.

Final answers

• (a) $P(\text{Yoga}) = \boxed{0.5}$, $P(\text{Yoga and Spin}) = \boxed{0.15}$
• (b) Not mutually exclusive, since $P(\text{Yoga and Spin}) = 0.15 \neq 0$.
• (c) Not independent, since $0.175 \neq 0.15$.
• (d) $P(\text{Spin} \mid \text{Yoga}) = \boxed{0.3}$