It rains in Spain once every 10 days, and when it does there is a 2% chance of hurricane in Hartford. When it does not rain in Spain, there is a 1% chance of hurricane in Hartford. What is the probability that it rains in spain when there is a hurricane in Hartford?

Question

It rains in Spain once every 10 days, and when it does there is a 2% chance of hurricane in Hartford.  When it does not rain in Spain, there is a 1% chance of hurricane in Hartford.  What is the probability that it rains in spain when there is a hurricane in Hartford?

ybarrap · Answer

Let RS = It rains in Spain (and NRS means no rain in Spain)
Let HH = There is a hurricane in Hartford

We know
$$
P(RS)=\cfrac{1}{10}\
P(NRS)=1-P(RS)=\cfrac{9}{10}\
P(HH|RS)=\cfrac{2}{100}\
P(HH|NRS)=\cfrac{1}{100}\
P(HH)=P(HH|RS)P(RS)+P(HH|NRS)P(NRS)\
=\cfrac{2}{100}\cfrac{1}{10}+\cfrac{1}{100}\cfrac{9}{10}
$$
What we need is the probability of RS given HH:
$$
P(RS|HH)=\cfrac{P(HH|RS)P(RS)}{P(HH)}
$$

And we know all three probabilities on the right.

Does this make sense?