# 4.2 Conditional Probability

## Watch Out For Crocodiles

This weekend it will rain with probability $$\frac{1}{4}$$, be sunny with probability $$\frac{1}{4}$$, and hail baseballs otherwise. If it rains, then there is a 50% chance that I will see a crocodile. If it is sunny, then crocodiles will hide themselves from me with probability $$\frac{3}{8}$$. If it hails, then I will see a crocodile with complete certainty.

What is the probability I see a crocodile this weekend? Please answer as a fraction of the form x/y.

Exercise 1

Using the Law of Total Probability, $$\Pr[\text{see a crocodile}] = \Pr[\text{see a crocodile}\;|\;\text{sunny}]\cdot\Pr[\text{sunny}]$$$$+\Pr[\text{see a crocodile}\;|\;\text{rain}]\cdot\Pr[\text{rain}]$$$$+\Pr[\text{see a crocodile}\;|\;\text{hail}]\cdot\Pr[\text{hail}]$$$$= \frac{5}{8}\cdot\frac{1}{4}+\frac{1}{2}\cdot\frac{1}{4}+1\cdot\frac{1}{2}=\frac{25}{32}$$