Question

In Vancouver, British Columbia, the probability of rain during a winter day is 0.42, for a spring day is 0.23, for a summer day is 0.16, and for a fall day is 0.51. Each of these seasons lasts one quarter of the year. If you were told that on a particular day it was raining in Vancouver, what would be the probability that this day would be a winter day?

Answer 1

Answer:

31.82% probability that this day would be a winter day

Step-by-step explanation:

We use the conditional probability formula to solve this question. It is

$P(B|A) = \frac{P(A \cap B)}{P(A)}$

In which

P(B|A) is the probability of event B happening, given that A happened.

$P(A \cap B)$ is the probability of both A and B happening.

P(A) is the probability of A happening

In this question:

Event A: Rain

Event B: Winter day

Probability of rain:

0.42 of 0.25(winter), 0.23 of 0.25(spring), 0.16 of 0.25(summer) or 0.51 of 0.25(fall).

So

$P(A) = 0.42*0.25 + 0.23*0.25 + 0.16*0.25 + 0.51*0.25 = 0.33$

Intersection:

Rain on a winter day, which is 0.42 of 0.25. So

$P(A \cap B) = 0.42*0.25 = 0.105$

If you were told that on a particular day it was raining in Vancouver, what would be the probability that this day would be a winter day?

$P(B|A) = \frac{0.105}{0.33} = 0.3182$

31.82% probability that this day would be a winter day