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
Asked by 10 months ago
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

1 Answer

Answered by 10 months ago
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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Asked: 10 months ago

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