All right, I am sure confused about probabilities.
Say that the chance of giving birth to a boy or girl is fifty-fifty. Say someone is expecting - it would be correct to say “There 50% chance that it will be a boy.”
Now suppose the wife already has three sons. What would be the correct thing to say?
The probability of you giving birth to a son is 50%, because your previous births are independent of this - ie, you giving birth to a son is independent of previous events.
or
The probability of having 4 sons is 0.5 x 0.5 x 0.5 x 0.5 or 0.0625. Hence there’s only a 6.25% that you will give birth to a son.
This is the correct one, all things being equal. However, ‘female’ sperm (ie. the sperms which cause girls) live longer but swim slower than ‘male’ sperm: this means that the probability doesn’t quite balance exactly for a given act of intercourse, as do other factors.
Assuming the gender of each child is determined completely at random (as opposed to there being some genetic factor that makes a person more likely to keep having boys) then 50% is correct. Here’s an explanation that might help:
Before having any kids, these are the possible ways it could go (B for boy, G for girl):
So, that’s 16 possibilities, and only one is BBBB. So the probability of having 4 boys is 1/16.
However, after having three boys, most of the possibilities I listed above have been eliminated. The only ones left are the ones that start with BBB, namely BBBG and BBBB. Thus, since there are two possibilities and one is BBBB, the probability of having 4 boys at this point is 1/2.
For the same reason, people who think a coin is “due” to come up tails if it has landed heads a bunch of times are simply wrong. Each new trial means a new 50/50 chance.
How many sons you already have has no effect on how many sons you will have in the future. The 50/50 means that if you have a sufficiently large number of births about half of them will be male. It doesn’t determine anything about any individual birth. Think of it this way - if you flip a coin 10 times and get 10 heads, the coin doesn’t compensate afterwards by giving more tails. Instead, it carries on getting roughly 50% heads and 50% tails and after a while this dilutes the effect of the 10 heads you got in a row.
50/50 chance if the coin is guaranteed to be a fair coin. It might be weighted, or two-headed, or simply being thrown by a very skilled flipper.
The same point applies to the chance of being born a boy or girl, I think. Over the population as a whole, the chances are 50-50, but I think there may be specific elements in the genes of a set of parents that weigh the odds towards one gender or another.
Of course a fair coin is required for this, and I am willing to admit that the boy/girl thing may not in fact be random. Still, the gambler’s fallacy is the easiest trap to fall into.
That is to say mistaking an independent variable for a dependent one. Simply put, coins (or wombs) ain’t got memory banks. So a coin has no way to know it is due to come up boy, and a womb has no way to know it is due to produce ‘heads.’
There are probably enough answers here already, but if you’re looking for a more intuitive explanation, I’ll give it a shot. Yes, the probability of having four sons is only 6.25%, but what’s the probability of having three sons? By the same kind of calculation, you can find that it’s 12.5%.
Now you know that your example person already has three sons, so you know she’s already in that 12.5%, and you’re asking what’s the probability that she will next be in the 6.25% with four sons. Well, it’s 50% obviously, right?
So there is much agreement that, assuming there’s no biological or genetic reason for there to be a bias toward boys, the chances of the next child being a boy are 50/50.
So, I think it’s time for some statistician to come along and tell us how to test the assumption. How many boys would a woman have to have before we could be, say, 95% confident that there is a biological or genetic reason why the couple keeps having boys?
Assuming, of course, that that’s all the kids she has. If she has 5 boys out of 50, well, then, maybe she doesn’t have a bias towards boys after all ;).
Well that is sort of vague, because “biological disposition” could refer to any arbitrary number. We can put it more succinctly if you like, for example, “How many boys in a row would a woman have to have before we were 95% confident that the true probability of her haveing a boy is greater than 66 2/3% (2 to 1)” or something like that. Unfortunately, my statistics is very rusty, but this is a simple question I am sure someone could answer.
No, ultrafilter is correct. You can set this up as a statistical test of hypothesis with the null hypothesis being the probability p of having a boy is .50. Now your sample shows 5 boys in a row. The P-value for this occurrence is around 3%. Since this is less than 5%, you can reject the null hypothesis and conclude, with 95% confidence, that p is larger than .50.
Suppose I am playing a game with two dice, and a success is considered as rolling two doubles.
Let say I’ll win a bet if I roll two doubles in a row, then is my chance of winning 1/6 or 1/36?
Say I roll a pair of dice and got a double, then my friend says to win the bet, I need to roll another double. So my chance of winning is 1/6. How is this consistent with case 1?
Getting doubles for both of two rolls? Your chances are 1/36.
It’s different because the probability of rolling a second double, given that you’ve already rolled one, is 1/6. You’re tossing aside the other cases where your first roll was a failure.
To put in terms of the children again… the probability that an arbitrary four-child family had all boys is 1/16. But the probability of a family of having a fourth boy, given that they’ve already had three, is 1/2 (stipulating as we are in this thread that the probability of any birth being a boy is 1/2).
The problem with this is that someone is going to go out, see a woman with 5 boys, and declare “I read there’s a 95% probability that there is a biological reason that someone with 5 boys keeps having boys.”
And that will be false. Because, in order to apply that test, you have to be talking about a specific woman without knowing already that she has five boys. If you’ve already scanned the population and selected those moms with five boys, the test doesn’t apply to that situation. That’s selection bias.
Just as an FYI IIRC in real world couples that have an existing string of 3-4+ boys or girls there is there is a substantial probability that single sex trend will continue well beyond the near 50/50 percentage normally seen in population demographics and birth probabilities. Why this is so is unknown, but is assumed to be biologically mediated beyond the realm of chance.
Re the OP Dr.MAth Boy or Girl: Two Interpretations Probability of Two Male Children
