Question

On rainy days, Joe is late to work with probability .3; on nonrainy days, he is late with probability .1. With probability .7, it will rain tomorrow.
(a) Find the probability that Joe is early tomorrow.

(b) Given that Joe was early, what is the conditional probability that it rained?

Answer 1

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

$P = 0.7*0.7 + 0.3*0.9 = 0.76$

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.

$0.7*0.7 = 0.49$

Early

From a), 0.76

Probability

$P = \frac{0.49}{0.76} = 0.6447$

64.47% conditional probability that it rained