You are thinking about whether to visit the zoo today. If you decide to take this trip, you will leave in the morning and return in the afternoon. However, you will cancel your trip if you believe that the chance of rain in the afternoon is higher than 45%. The historical data suggests that 1) when it is cloudy in the morning, 60% of days rain in the afternoon; 2) Cloudy mornings are common (about 45% of days start cloudy); and 3) The current month is a dry month with 30% of days raining in the afternoon. If it is not cloudy this morning, what is your estimated probability of rain in the afternoon?
Question
You are thinking about whether to visit the zoo today. If you decide to take this trip, you will leave in the morning and return in the afternoon. However, you will cancel your trip if you believe that the chance of rain in the afternoon is higher than 45%. The historical data suggests that 1) when it is cloudy in the morning, 60% of days rain in the afternoon; 2) Cloudy mornings are common (about 45% of days start cloudy); and 3) The current month is a dry month with 30% of days raining in the afternoon.
If it is not cloudy this morning, what is your estimated probability of rain in the afternoon?
Solution
To solve this problem, we need to use the concept of conditional probability.
First, let's denote the events as follows:
C = It is cloudy in the morning R = It rains in the afternoon
We are given the following probabilities:
P(R|C) = 0.60 (probability it rains in the afternoon given it is cloudy in the morning) P(C) = 0.45 (probability it is cloudy in the morning) P(R) = 0.30 (probability it rains in the afternoon)
We are asked to find P(R|~C), the probability it rains in the afternoon given it is not cloudy in the morning.
We know that P(R) = P(R and C) + P(R and ~C), which can be rewritten as P(R) = P(R|C)P(C) + P(R|~C)P(~C).
We can solve this equation for P(R|~C) to find:
P(R|~C) = (P(R) - P(R|C)P(C)) / P(~C)
Substituting the given values:
P(R|~C) = (0.30 - 0.60*0.45) / (1 - 0.45)
P(R|~C) = (0.30 - 0.27) / 0.55
P(R|~C) = 0.03 / 0.55
P(R|~C) = 0.0545 or 5.45%
So, if it is not cloudy in the morning, the estimated probability of rain in the afternoon is 5.45%.
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