Given identical twin males, Michael and Douglas, Douglas and his wife have four children, all of whom are girls. Michael and his wife have only one child, a boy, and are expecting another. What are the chances it will be another boy? If Michael and his wife have more children, what are the chances they will all be boys? Is there any support/evidence for, or a study about, one twin of an identical pair will have children all of the same gender and the other twin will have children all of the same (or opposite) gender?
Note: This is based on literature searches I've done a while ago out of general curiosity. I'm in no way an expert on human reproduction.
If I understand you correctly you are basically asking if there are any evidence for sex biases in offspring between families, that has a heritable genetic basis. This has been an active research topic for a long time and there are a number of proposed mechanisms for how this can be achieved. Historical data sometimes shows deviations in sex ratios over time, e.g. in 20th centrury Europe after WWI and WWII (see data in Gellatly, 2009). There are also other reported/hypothesized deviations in sex ratio e.g. trends in sex ratio due to paternal age (e.g. Jacobsen et al, 1999), sex biases of offspring due to earlier insemination in the menstrual cycle, and the child rank. However, from what I've seen, the evidence hasn't been conclusive and different studies point in different directions.
As Irving et al (1999) point out, possible mechanisms for sex ratio biases can be:
- deviations in X/Y-ratio of sperm
- selection of sperm within the female reproductive tract
- biases in implantation success and/or survival for embryos of different sex
Most evidence seems to point against explanation one, see tests in e.g. Irving et al (1999) and Graffelman et al (1999). Graffelman et al (1999) also tests for an effect of male age, but do not find support for this idea.
Another interesting study is Curtsinger et al (1983), which tries to look at vertical population transmission of sex ratio variation in Japan. They conclude that there is a marginally significant effect on the paternal side, which would support the idea that sex biases might run in families.
An analysis of vertical transmission of sex-ratio modifying factors that excludes effects of birth order in both the parental and offspring generations has detected a marginally significant paternal effect.
However, they also conclude that the genetic variability in sex ratio is very small, if there is one. This study also does not find support for effects of e.g. paternal age or child birth order.
Gellatly (2009) provides a model of how an autocorrelation in sex ratio between generations could be explaned (while still having a stable population-level sex ratio), using an autosomal gene modifying male sex ratio. Note thought that this is a proposed model, and not empirical evidence. However, Gellatly (2009) also use genealogical/pedigree data from North America and Europe to show that there is an heritable component in the human sex ratio (an association between sex ratio in F1 and F2 generations). However, from what I can see the evidence is weak, and the models only explains 0.5% of the variance in the data. The introduction to this paper also has a useful overview of the subject with more references you might want to check out.
All in all, my perception is that there are some evidence that certain sexes can be overrepresented in families, which could potentially mean that the sex ratio in the offspring of a twin brother can contain information on the probablility that his brother will have a son or daughter. However, to me, the evidence is far from conclusive and the estimated effects seem rather weak. So as a null hypothesis, you should definately go for a 50/50 chance of a son/daughter, irrespectively of what the sex ratio of the twin brother is.
As for the general calculation of probabilities of different scenarios, you just use the product of probabilities and sometimes summation of birth orders. For instance (if assuming 50/50 sex ratio), the probability of having three sons is $0.5^3= 12.5\%$ (births: MMM) and the probability of having 2 sons and one daughter is $3*0.5^3= 37.5\%$ (births: FMM, MFM or MMF, i.e. three ways to achieve this scenario out of eight).
The question you're asking is essentially a version of an old riddle:
If you flip a coin ten times and it comes up heads each time, what are the odds that the coin will come up heads when you flip it for an eleventh time?
And of course the answer is that the odds are fifty-fifty. You could instead argue that 10 heads in a row is evidence that the coin is weighted somehow, and the way you would deal with that is by flipping that same coin 1000 more times, or (somewhat less reassuringly) by finding 100 coins identical to your first one and then flipping them each 10 times.
It's the same thing with children's genders. For every family with 6 girls there are hundreds more families with more mixed genders. The only effect known to skew the male-female sex ratio at birth significantly is wide scale sex-selective abortion (it tends to reduce the number of females).