A First Course In Probability, 9th Edition, Indian Version, Sheldon Ross, Chapter 3, Problems, 33.

Question

The meteorology department predicts the rain correctly with a probability of 4/5. When rain is predicted Amit travels by his car. When rain is not forecasted, he travels with his car with a probability of 0.5. Assuming that it rains in a day with a probability of 1/2. Find the probability that Amit does not travel by car given it rains.

Answer 1

In above figure,

• $R$ means “It rains” and $\overline R$ means “It doesn’t rain.
• $P$ means “It is predicted that today will rain” and $\overline P$ means “It is  not predicted that today will rain”.
• $C$ means “Travels by car” and $\overline C$ means “Doesn’t travel by car”.

$\therefore P(\overline C | R) = \frac{P(\overline C \cap R)}{P(R)} = \frac{\frac{1}{2} * \frac{4}{5} * 0 + \frac{1}{2} * \frac{1}{5} * 0.5}{\frac{1}{2}} = 0.1$