For numerical calculations, your method doesn't ever really break down and, moreover, Brennan is essentially doing the same thing you are, but with a sample of 16 people instead of 100 to make the math simple enough to do mentally.
A more Bayes-theorem styled calculation tells us that we have 1:3 odds initially (as there are 1/4 men and 3/4 women) and the Virtuist evidence updates it by a factor of 2:3 (as Virtuists are 2/4 of men and 3/4 of women), so we end with 2:9 odds. I think this is easier than what either you or Brennan are doing, but it's a matter of taste and of what's more intuitive. (Which certainly varies from person to person; I find two-elevenths easier to grasp than 22.2%)
Doing Bayesian calculations formally is more important where you are doing symbolic calculations, especially with continuous probability distributions.
Edit: Also, 22.2% is wrong, which I didn't realize at first; it's 2/9, not 2/11. You want to compute 12.5/(12.5+56.25) instead.
You want to compute 12.5/(12.5+56.25) instead.
Of course, I don't know how I missed that...
Now on to Monthy Hall the linked explanation is not that intuitive to me.
To me the intuitive explanation is that if I chose a goat, switching gives me 100% possibility to get a car and not switching 0%, if I chose a car, switching gives me 0% possibility and switching 100%, thus my original 2/3 chance to win a goat wins me a car with the same 2/3 chance if I switch.
I don't know if it is Bayesian what I am doing... let's play with 4 doors, 3 goats 1 cup, er, car. ...
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
Notes for future OT posters:
1. Please add the 'open_thread' tag.
2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)
3. Open Threads should be posted in Discussion, and not Main.
4. Open Threads should start on Monday, and end on Sunday.