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Answer:
a) 76% probability that Joe is early tomorrow.
b) 64.47% conditional probability that it rained
Step-by-step explanation:
We have these following probabilities:
A 70% probability that it will rain tomorrow.
A 30% probability that it does not rain tomorrow.
If it rains, a 30% probability that Joe is late and a 100-30 = 70% probability that Joe is early.
if it does not rain, a 10% probability that Joe is late and a 100-10 = 90% probability that Joe is early.
(a) Find the probability that Joe is early tomorrow.
Either it rains(70% probability) and he is early(70% probability when it rains), or it does not rain(30% probability) and he is early(90% probability when it does not rain). So
76% probability that Joe is early tomorrow.
(b) Given that Joe was early, what is the conditional probability that it rained?
By the Bayes theorem, this probability is:
The probability that it rained and he was early divided by the probability he was early.
Rained and early
70% probability it rains.
70% probability he is early when it rains.
Early
From a), 0.76
Probability
64.47% conditional probability that it rained