Question

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?)

Answer 1

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.
`Pr(n)=1/20," "n=1,2,...,20.`

Let `X` be the number of darts (out of 20) that hit `n`. Then
`X" "~" " "Bin"(20, 1/20)`.

Therefore, the probability that 8 darts hit `n` is
`Pr(X=8)=((20),(8))(1/20)^8(19/20)^12," "`
`color(white)(Pr(X=8))~~ 2.659 xx 10^"-6"," "n=1,2,...,20.`

To find the probability that 8 darts hit the same number, for all numbers `n=1,2,...,20,` just multiply `Pr(X=8)` by 20.

`Pr("8 darts hit one area")=8 * Pr(X=8)`
`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%.`