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?

1 Answer
Nov 2, 2015

Use the Binomial distribution


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