Question

The chance of rain on a day when an outdoor picnic is arranged is \(40\%\). If it rains, the chance that the picnic will be ‘good and enjoyable’ is \(30\%\). However, if it does not rain, the chance that the picnic will be ‘good and enjoyable’ is \(80\%\). The chance (in %) that it was raining on the day of picnic, given that the picnic was indeed ‘good and enjoyable’ is (in integer).

Hint:

Bayes’ theorem helps in finding conditional probabilities in reverse order: \[ P(A|B)=\frac{P(B|A)P(A)}{P(B)} \] Always calculate the total probability of the given condition first.

Answer 1

Correct Answer: 20

Step 1: Name the events.
Let $R$ be rain and $G$ be a good picnic. We are given $P(R)=0.4$, so $P(R^c)=0.6$, with $P(G\mid R)=0.3$ and $P(G\mid R^c)=0.8$. We want $P(R\mid G)$.

Step 2: Find the chance of a good picnic.
By total probability,
\[ P(G)=0.3(0.4)+0.8(0.6)=0.12+0.48=0.60 \]

Step 3: Apply Bayes' rule.
\[ P(R\mid G)=\frac{P(G\mid R)P(R)}{P(G)}=\frac{0.12}{0.60}=0.2 \]

Step 4: As a percent.
That is $20\%$.
\[ \boxed{20\%} \]