# 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?

Nov 2, 2015

Use the Binomial distribution

#### Explanation:

A fair die, you expect $\frac{1}{6}$ of the trials to come up with a value of 1. So, the expected number of ones with 100 trials is $\frac{100}{6} \approx 17$

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

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

$P \left(x = 5\right) = 0.00029$

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