The following is an exercise from the classical textbook of Feller on probability theory.
Four girls take turns at washing dishes. Out of the total of four breakages, three were caused by the youngest girl. Was she justified in attributing the frequency of her breakages to chance?
This is a variant of the balls and bins problem and we are asked to compute the probability of 3, out of 4 indistinguishable balls, ending up in a specific bin, which I think is given by
Feller seems to disagree, however. The textbook solution to this problem is 13256, without any further explanation though. It’s not at all obvious to me where the 13 comes from, I have to say. Could this be a typo? Or perhaps I am missing something in the problem?
There is a tradition in frequentist statistics to measure the significance of a result by how small the probability is of observing by chance that result or an equally extreme or more extreme result.
The youngest sister breaking all four items is certainly more extreme than her breaking three, and this is probably why its probability was included in the calculation of what is effectively a one-tailed test, adding up all the probability in the tail to give 13256.
What is less clear is whether the oldest sister breaking three items is as extreme; if so, and similarly with the other sisters, then the answer would be 52256, which is starting not to look so significant. The youngest sister should demand a four-tailed test.