Learn / Library / Expected value

Venues left empty

Expected valueMedium

$6$ orders are each sent independently and uniformly at random to one of $3$ venues. What is the expected number of venues that receive no order? Give a fraction in lowest terms.

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Use one indicator for each venue being empty.

Solution

A given venue is missed by all the orders with chance $(\frac{2}{3})^{6} = \frac{64}{729}$ .
By linearity over the venues, the expected number is $3 \times \frac{64}{729} = \frac{64}{243}$ .

Answer: $\frac{64}{243}$

Next question Back to the library