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

Answered by Aaron 10 months ago

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