Question

Alexis rolls an 8-sided die 24 times. What is the probability a "five" is rolled at least 4 times, but no more than 18 times?

Answer 1

`~~0.3524`

Explanation:

The binomial probability relation is:

`sum_(k=0)^(n)C_(n,k)(p)^k(1-p)^(n-k)=1`

Here we have `n=24, p=1/8` and we're looking at `4<=k<=18`. We can sum up to that or we can look at `1-(0<=k<=3+19<=k<=24)`. Both involve a lot of work so I'll use Google Sheets and find the cumulative binomial distribution up to 18 and then subtract the one up to 3.

https://docs.google.com/spreadsheets/d/1jwBwig9fm-r-7F4nQSlk972MtMl13tRCMCwkrJGIL10/edit#gid=0

It turns out that at the cumulative probability, summed to 18 successes, we're so close to 1 that it's roughly a dozen 9's immediately after the decimal.

The cumulative probability at the 3 successes mark is roughly 0.6476.

Therefore, our probability is roughly 0.3524.