Does knowing the sex of each of a couple’s first N children have any marginal value in predicting the sex of child N+1?
For those who are interested in sex ratio bias itself, the enWikipedia article is pretty good.

As for predictive power, a Bayesian might tell you that by knowing a couple can and did have a male/female should improve, if only slightly, your belief that their next would be the same.  It's by no means predictive and probably largely indiscernible from random chance, but hey it's something!

More to the point, at the very least you might as well just guess their next child will be a son.  As the link above will tell you, the human sex ratio at birth has varied but in general males are more likely to be born - somewhere around 1.03 to 1.08 males per female.  Here's another in-depth study, from 2005, dealing with the last few decades in the US.

That study has Figure 3 (below) and Table 3, which show the raw birth ratio over successive births, and the data are pretty clear: successive births reduce bias.



Those numbers are way more clear than I thought they would - just watch those means move!  I wish there were more information on the variances - clearly there's more down by the 8th child - but that's pretty solid.  Here's another study from Denmark in 1999 doing some of the same stuff (Denmark was popular for this work).  They include the absolutely fascinating Table 2 (below), which breaks down the odds of giving birth to a male given whatever came before.  Which, in the end, is as predictive as you can get, right?



Basically, it's sort of what you'd expect: the more females you have, the more likely you are to have another female; the more males, the more likely the next is a male.  They used about 1.4 million kids but there are some interesting oddities in that table at the extremes.  The best chance for a male?  Your fifth kid, following FMFM.  Best for a female?  Your fifth, after MMMF.

There are some definite issues that can twist the data, particularly when families have another kid to try to get a child of a specific sex.  If you can tease out some of those environmental factors, you'd be better at guessing; still, I wouldn't bet the house on it.  Memorize that table but, when in doubt:


Guess male
Lotta females first?  Maybe guess another one.
A bunch of males?  Guess another male.