Half (50%) of all rainy days start out cloudy. 40% of all days start out cloudy. It ends up raining only 10% of the time. If the morning is cloudy today, what is the chance that it will rain today?8%12.5%2%80%25%
Question
Half (50%) of all rainy days start out cloudy. 40% of all days start out cloudy. It ends up raining only 10% of the time. If the morning is cloudy today, what is the chance that it will rain today?8%12.5%2%80%25%
Solution
The problem can be solved using Bayes' theorem, which is a way of finding a probability when we know certain other probabilities. The formula is:
P(A|B) = P(B|A) * P(A) / P(B)
Where:
- P(A|B) is the probability of event A given event B is true
- P(B|A) is the probability of event B given event A is true
- P(A) and P(B) are the probabilities of events A and B respectively
In this case, we want to find out the probability that it will rain given that the morning is cloudy (P(Rain|Cloudy)). We know the following:
- P(Cloudy|Rain) = 0.5 (50% of all rainy days start out cloudy)
- P(Cloudy) = 0.4 (40% of all days start out cloudy)
- P(Rain) = 0.1 (It ends up raining only 10% of the time)
Substituting these values into the Bayes' theorem formula gives:
P(Rain|Cloudy) = P(Cloudy|Rain) * P(Rain) / P(Cloudy) = 0.5 * 0.1 / 0.4 = 0.125 or 12.5%
So, if the morning is cloudy, there is a 12.5% chance that it will rain.
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