Question

On any given day in which it had rained, there was a 40% chance that the morning was cloudy. When you wake up in the morning, you notice that it is cloudy (event C). If you want to know the probability that it will rain (event R), which formula would you use?

a.) P(R ∣ C) = (PCR)P(R)/(PC)
b.) P(R ∣ C) = (PCRPC)/P(R)
c.) P(C ∣ R) = (P(R∣C)P(R)P(C)
d.) P(C ∣ R) = P(R∣C)PC)/P(R)

Answer 1

To find the probability of rain given that it is cloudy, you use the conditional probability formula, with option a.) P(R | C) = P(C AND R)/P(C) being the correct one.

Step-by-step explanation:

When you want to find the probability that it will rain given that it is cloudy, you need to use the formula for conditional probability.

The correct formula to apply in this situation is option a.), which is written as P(R | C) = P(C AND R)P(C)/P(R). This formula is derived from the definition of conditional probability where P(C AND R) is the probability of both events happening together, P(C) is the probability of it being cloudy, and P(R) is the probability of rain.

Based on the standard formula of conditional probability P(A|B) = P(A AND B)/P(B), where P(B) is greater than zero, we can determine that of the given options, a.) is the only formula structured properly to give us the probability of rain given that it is cloudy, making it the correct answer.