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, \(\Pr[\text{see a crocodile}] = \Pr[\text{see a crocodile}\;|\;\text{sunny}]\cdot\Pr[\text{sunny}]\)\(+\Pr[\text{see a crocodile}\;|\;\text{rain}]\cdot\Pr[\text{rain}]\)\(+\Pr[\text{see a crocodile}\;|\;\text{hail}]\cdot\Pr[\text{hail}]\)\( = \frac{5}{8}\cdot\frac{1}{4}+\frac{1}{2}\cdot\frac{1}{4}+1\cdot\frac{1}{2}=\frac{25}{32}\)