Suppose a woman has a brother with an X-linked recessive disease (e.g., Duchenne Muscular Dystrophy), neither of her parents has the disease expressed (hence her mother is a recessive carrier), and she gives birth to a normal son. What is the probability that she's a carrier? The woman starts with a 1/2 probability of having inherited the bad X chromosome from her mother (receiving one X chromosome from mother and one from father). Having given birth to a single healthy son, is her probability of being a carrier still 1/2?

A woman who is a carrier (C) has one bad X chromosome and hence has a probability of 1/2 of giving birth to a healthy son (who receives one of those two X chromosomes), hence has an overall probability of 1/4 that she's a carrier and gives birth to a healthy son. If she's not a carrier, then she always gives birth to a healthy son, and hence has an overall probability of 1/2 that she's not a carrier and gives birth to a healthy son. But after giving birth to the healthy son, the probability that she's a carrier drops to 1/3 (because it's twice as likely, 1/2 compared to 1/4, that she'd have a healthy son if she's not a carrier), or more strictly speaking the conditional probability p(C | 1 h.s.) = 1/3, where C= carrier and h.s.=healthy son.

If she has n healthy sons in a row, we can write
p(C | n h.s.) = p(n h.s. | C) p(C) / p(n h.s.),
where p(n h.s.) = p(n h.s. | C) p(C) + p(n h.s. | NC) p(NC) = (1/2)^n (1/2) + 1/2 .
Hence p(C | n h.s.) = (1/2)^n (1/2) / ((1/2)^n (1/2) + 1/2) = 1 / (1 + 2^n)
(and equals 1/3 for n=1, as above).

In an alternate "frequentist" approach, the original 1/2 probability is treated like a coin flipper, so that no matter how many healthy sons she has, someone would say the probability is still 1/2 that she's a carrier, and hence still probability 1/4 that the next son will have the disease, because each time her state is determined by flipping an unbiased coin and there's no notion of the hidden state (carrier vs. non-carrier, with different probabilities for diseased offspring). But one should instead have the intuition that after having some number of healthy sons in a row, the likelihood that she's a carrier must be dropping. (Whereas if she has even a single diseased son at any point, the probability that she's a carrier jumps to 1.)

She doesn't flip a coin before each birth to decide whether or not she's a carrier, she's either a carrier or not, and we try to infer that probability based on accumulated evidence. This problem is instead the problem of the casino with some probability of choosing a loaded die (also a "hidden state"), and inferring the posterior probability of having chosen a loaded die, based on rolling some number of consecutive sixes.

(Apparently this is not always well taught in genetics courses.)

This is different from the problem of a couple having children (in the simplest model), in which the sex of each successive one is determined by independent coin flips, so that a couple with three daughters still has a 1/2 probability that the fourth will be a son. (Unless there is as well some hidden state as well in this problem, say some known biological condition with some a priori probability that causes male children to be miscarried -- then the likelihood of the couple being subject to this condition is increased the more successive daughters they have, and for some sufficiently large number it becomes more likely than not. And actually the ratio of female:male births is not exactly 1:1 for a variety of reasons.)