# What is the probability that if you threw 20 darts, that 8 of them would hit the same area?

## Assume you hit the dart board every time, and that you had equal chance of hitting the different numbers on the board (e.g 1/5 chance?)

Jan 25, 2017

The probability is approximately 0.00213%.

#### Explanation:

If we have an equal chance of hitting all the numbers on the board, and we are guaranteed that we hit the board, then each number has a 1 in 20 chance of being hit.
$P r \left(n\right) = \frac{1}{20} , \text{ } n = 1 , 2 , \ldots , 20.$

Let $X$ be the number of darts (out of 20) that hit $n$. Then
$X \text{ "~" " "Bin} \left(20 , \frac{1}{20}\right)$.

Therefore, the probability that 8 darts hit $n$ is
$P r \left(X = 8\right) = \left(\begin{matrix}20 \\ 8\end{matrix}\right) {\left(\frac{1}{20}\right)}^{8} {\left(\frac{19}{20}\right)}^{12} , \text{ }$
$\textcolor{w h i t e}{P r \left(X = 8\right)} \approx 2.659 \times {10}^{\text{-6"," }} n = 1 , 2 , \ldots , 20.$

To find the probability that 8 darts hit the same number, for all numbers $n = 1 , 2 , \ldots , 20 ,$ just multiply $P r \left(X = 8\right)$ by 20.

$P r \left(\text{8 darts hit one area}\right) = 8 \cdot P r \left(X = 8\right)$
color(white)(Pr("8 darts hit one area"))~~ 8 * 2.659xx10^"-6"
color(white)(Pr("8 darts hit one area"))~~ 2.13 xx 10^"-5 "=" "0.00213%.