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What is the probability of the other child being a girl?
Wikipedia: Boy or girl paradox
Reddit: I don't get it
Youtube: Internet baffled by probability riddle
This is what inspired me to stick with it: Youtube: Feynman inspiration
Let P be the probability of at least one of two children being a girl.
Mary has two children, if that is all you know, these families are equally probable: BB BG GB GG. Thus P = 3/4.
If Mary tells you she has at least one boy, these are equally likely: BB BG GB, so P = 2/3.
If Mary tells you her first child is boy, these are equally likely: BB BG, so P is 1/2.
Suppose the culture is that, at a certain hour, mothers flip a coin to choose a child to take out for a walk. You will see half the children are boys, half are girls. You see Mary with a boy. So Mary's family must be one of: BB BG GB. But the probability of choosing a B to walk with is TWICE if BB. BB BG GB are not equally probable, BB is twice as probable, if all you observe is the walk. P=1/2.
But if Mary told you, prior to you observing the walk, that she has at least one boy, P=2/3. What if you subsequently observe a walk with a girl? P=1, obviously.
What if you subsequently observe Mary walking a boy? P=1/2 or P=2/3? I say P=1/2. Mary's prior statement is a corollary of this new observation.
Consider the culture that a boy born on a Tuesday is considered special. These boys brag about being a “Tuesday Boy”.
You observe Mary walking a boy. P=1/2. The boy then says to you one of:
- “the Pope is Catholic”
- “I am Catholic”
- “I am a Tuesday Boy”
P does not change, P=1/2.
But what if the boy says “I am THE Tuesday Boy”? This information excludes the possibility of the sibling being another Tuesday Boy. Instead of 14 possibilities for the sibling, there are now 13 equally probable siblings, and 7 are girls. P=7/13 = .538
Here is where Presh Talwaker makes his mistake. The grid isn't helpful if the not all cases have equal probability. Presh says “there are 27 check marks”. No, $B_2B_2$ should have two check marks, just as in the classic “I am a Catholic” statement, $BB$ would have two check marks, otherwise you would not get P=.5. In Presh's grid, 14 still have a girl. So he should have said P=14/28=.5. “and that's the answer!”. Presh doesn't realize he is doing the “I am A Tuesday Boy” case. If he wants to analyze the “I am THE Tuesday Boy” result, he needs to drop two check marks, and get P=14/26=7/13=.538.