This question stems from the boy-girl paradox, but it is not a repeat question so please read on.
The boy-girl paradox question is worded in many different ways, but in essence it says:
"If a family has 2 children, given that at least 1 child is a boy, what is the probability that the other child is also a boy."
The normal "correct" answer to this question is calculated by creating a sample space S of possible outcomes as:
S = { (b, b), (b, g), (g, b), (g, g) }
Let event A = (b, b) => P(A) = 1/4
Let event B = "at least 1 child is a boy" = { (b, b), (b, g), (g, b) } => P(B) = 3/4
Thus concluding that P( A | B ) = P(AB) / P(B) = (1/4)/(3/4) = 1/3
This is where my confusion begins, because the answer above is necessarily saying that gender probability is dependent on another person's gender. I will try to emphasize my point by starting with questions that can only be answered with boolean values. This is trivial, but please bear with me.
True or False: Any person's biological gender is independent of any other person's biological gender. (TRUE)
True or False: The probability of any person born male is 1/2. (TRUE)
True or False: The probability of any person born female is 1/2. (TRUE)
Since we are sure that a person's gender is independent of any previous person's gender, we can clearly see that the probability of 1 person having a specific gender is 1/2, even if the last 5 million people before that person were all the same gender.
This is the same as flipping a coin. If I flip 5 million heads in a row, the fact that this happened obviously has zero affect on the outcome of the next flip of the coin, because outcomes of coin tosses are independent events. We all know and accept this, it is intuitive and obvious.
My question:
How can an event be simultaneously independent and dependent? You may say "they can't!", but this would contradict the accepted conclusion to the boy-girl question.
If an event can never be simultaneously independent and dependent, then that means the event's probability will never be influenced by any number of that same event's previous outcomes.
And in the boy-girl problem, the answer necessarily implies that the probability of some child's gender depends on information about some other child's gender! Hence the different probability of 1/3 instead of 1/2.
I fundamentally disagree that the accepted sample space is actually correct, because it necessarily implies that an independent event can sometimes be dependent. I believe the true sample space is: { (b,b), (g,g), (b,g) }.
If I asked you the following question with no additional information:
"What is the probability that this coin I will flip will land on heads?" You would have to answer: "The probability is 1/2". Now I tell you that I have more information about the coin tosses before this, and you would have to say: "It doesn't matter! Coin tosses are independent!". So why is coin-toss independence any different from other events independence?
If it is actually true that the boy-girl problem is 1/3, wouldn't it logically follow that the outcome of a coin-toss can sometimes be something other than 1/2?
The boy-girl question is not related to independent coin tosses, as you suppose above. It is related to a fixed set of outcomes of an experiment and partial information about that set.
You need to compare it to the following. I have observed 2 coin tosses (already completed) and I know one of the tosses is a head. What is the probability that the other one is a head as well? The fact that one is a head means that 2 tails is impossible, hence the answer.