# Probability with Replacement

Probability with Replacement is used for questions where the outcomes are returned to the sample space again. This means that once the item is selected, it is replaced in the sample space, so the number of elements of the sample space remains unchanged.

## Example

A jar contains five balls numbered $1, 2, 3, 4$ and $5$. A ball is chosen at random, and its number is recorded. The ball is then returned to the jar. This is done a total of five times.

(a)    Find the probability that each ball is selected exactly once.

\begin{aligned} \displaystyle &=\frac{5}{5} \times\frac{4}{5} \times\frac{3}{5} \times\frac{2}{5} \times\frac{1}{5} \\ &= \frac{4!}{5^4} \end{aligned}

(b)    Find the probability that at least one ball is not selected.

\begin{aligned} \displaystyle &= 1-\frac{4!}{5^4} \end{aligned}

(c)    Find the probability that exactly one of the balls is not selected.

\begin{aligned} \displaystyle \Pr(\text{Ball 1 is not selected and all the rest at least once}) &= \frac{4}{5} \times \frac{4}{5} \times \frac{3}{5} \times \frac{2}{5} \times \frac{1}{5} \\ &= 4 \times \frac{4!}{5^5} \\ \Pr(\text{Exactly one not selected}) &= 5 \times 4 \times \frac{4!}{5^5} \\ &= 4 \times \frac{4!}{5^4} \end{aligned}