The Emory Harrison family of Tennessee had 13 boys. What is the probability of a 13-child family having 13 boys?

Consider a random experiment with only two possible outcomes (it's called Bernoulli experiment). In our case the experiment is the birth of a child by a woman, and two outcomes are "boy" with probability #p# or "girl" with probability #1-p# (the sum of probabilities must be equal to #1#).

When two identical experiments are repeated in a row independently from each other, the set of possible outcomes is expanding. Now there are four of them: "boy/boy", "boy/girl", "girl/boy" and "girl/girl". The corresponding probabilities are:
P("boy/boy") #= p * p#
P("boy/girl") #= p * (1-p)#
P("girl/boy") #= (1-p) * p#
P("girl/girl") #= (1-p) * (1-p)#
Notice that the sum of all above probabilities equals to #1#, as it should.
In particular, probability of "boy/boy" is #p^2#.

Analogously, there are #2^N# outcomes of #N# experiments in a row with the probability #N# "boy" results equal to #p^N#.

For detailed information on Bernoulli experiments we can recommend to study this material on UNIZOR by following links to Probability - Binary Distributions - Bernoulli.