My three children were given three rabbits from friends this easter. The rabbits were an "accident", and so that this doesn't happen again, we will have the male rabbits castrated. My daughter is very anxious about this. She pleaded with me not to have the procedure done. I told her that we would definitely castrate the males, unless the group consisted only of females or only of males. (Determining the sex with young rabbits is a bit tricky).
"How likely is it, that they all have the same sex when we go to the veterinarian?"
My answer went as follows:
Solution #1:
If we symbolize a male rabbit with a "1" and a female rabbit with a "0", the list of all possibilities is:
1.) 000 2.) 001 3.) 010 4.) 011 5.) 100 6.) 101 7.) 110 8.) 111
Assuming that the likelihood of all the possibilites are the same, the chance of no castration taking place is 2/8 = 1/4
My daughter disagreed, and was much more optimistic:
Solution #2:
There are only two sexes, so two rabbits will always have the same sex with absolute certainty. The probability of the third rabbit having that same sex is 50%. So the probability is 1/2.
What IS the probability of three rabbits having the same sex?
Agreeing with / expanding upon @Lulu's comment: solution 1 is correct, and solution 2 is incorrect.
The subtle mistake made in solution two is: the sex of the third rabbit is not independent of the other pair, precisely because the way that the divisions themselves are made depends on the sexes of the rabbits. Look carefully again at your (correct) list of 8 outcomes in solution 1, and you can see that the actual probability of agreement given that algorithm is in fact $1/4$: If you are in any of the middle 6 situations, the third rabbit will certainly not have the same sex as the other two. If you are in any of the outermost situations, the third rabbit will certainly have the same sex as the other two.