##### Example on Expectation

A school class of 120 students is driven in 3 buses to a symphonic performance. There are 36 students in one of the buses, 40 in another, and 44 in the third bus. When the buses arrive, one of the 120 students is randomly chosen. Let X denote the number of students on the bus of that randomly chosen student, and find E[X].

Since the randomly chosen student is equally likely to be any of the 120 students, it follows that:

$P \{ X = 36 \} = \frac{36}{120} \\ P \{ X = 40 \} = \frac{40}{120} \\ P \{ X = 44\} = \frac{44}{120}$

Hence,

$E[X] = 36 (\frac{3}{10}) + 40(\frac{1}{3}) + 44(\frac{11}{30}) = \frac{1208}{30} = 40.2267$