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.

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.

J
240 points

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?

conditional probability
bayes theorem

8.7k points

let,

*R * be the event that it rains *E * be the event that Joe is early

Find the probability that Joe is early tomorrow

$$P(E)=P(E|R)P(R)+P(E|R^c)P(R^c)=(1-0.3)(0.7)+(1-0.1)(1-0.7)=0.76$$

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

Using Bayes formula,

$$P(R|E)=\frac{P(RE)}{P(E)}=\frac{P(E|R)P(R)}{P(E)}=\frac{(1-0.3)(0.7)}{(0.76)}\approx 0.6447$$

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