Question

If you roll a die `100` times and get `15` ones, what is the probability that the die is not fair? What about if you get `5` ones?

Answer 1

Use the Binomial distribution

Explanation:

A fair die, you expect `1/6` of the trials to come up with a value of 1. So, the expected number of ones with 100 trials is `100/6~~17`

Getting 15 ones is not too far off from 17, so that would not be an unusual occurrence (`P(x=15) = 0.10`

However, a result of only 5 ones would be quite rare ...

`P(x=5) = 0.00029`

The problem did not state any hypothesis testing . However, if one assumes say an alpha = 0.05 with a left tail test, then for a Binomial with 100 trials and p = 1/6, the critical value = 10. That is, a roll of 10 "ones" or less has a probability of 4.3% which is as close as we can get to an alpha of 5% with a "discrete" distribution. Since, the value of 5 ones falls within the critical area , one can conclude that the die is not fair.

Hope that helps