Question

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

Answer 1

If the probability of giving birth a boy is `p`, then the probability to have `N` boys in a row is `p^N`.
For `p=1/2` and `N=13`, it is `(1/2)^13`

Explanation:

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.