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Sometimes people never learn. Jacque Wilson, in a piece for CNN titled: Calculating the odds: 12 sons in a row reports the odds at about 1 in 4000.
The problem first is in calculating the odds after discovering an instance of the event. I can’t just flip a coin 12 times, get 12 heads and say the odds of that are one in 4000. Any sequence of flips – heads and tails – is just as likely to happen. We only focus on twelve heads because it appears unique to us. Look at 4000 such sequences and one sequence of 12 heads is likely to happen. The actual odds are (10.5*(*12))**4000 of not having an incidence of 12 heads in the 4000 tries. 10.5**12 is the odds of it not happening in a single trial. That is very high at 0.9998. but after 4000 trials the odds of not seeing it happen is about 0.37.
With the situation with the twelve sons, the second exponent is not 4000, but the number of women with 12 children.
But it does not stop there as the story could have been written had the reported discovered any woman with a large number of children of the same sex. If the subject of this story come to her attention when she had ten sons she could have written a similar story. With all the possible ways the story could have been generated it is impossible to calculate the exact probability.
And the reporter commits an even graver error. She goes on to note that child number 13 is due in May and then goes on to say the odds of having a 13th son is 1 in 8000. I’m sorry the odds are very close to 1 in 2. The other 12 have already arrived – they have no real influence on the odds of 13th child being a boy or a girl.
Actually the odds may be even closer to 1 than you’ve said. You’ve aimed that the odds for the birth of a boy is 0.5. This may be to for the population in general but in this case given the fact of so many boys born it is possible there is some medical reason why this woman is predisposed to giving birth to boys.
You assumed the gender of each child is independent of the gender of the previous children. This may not be the case.
Berry,
Yes 4096 is the exact number if the 0.5 is correct. I just rounded it to 4000 – close enough. And yes that is the number the reporter is looking for. Problem is that it is the wrong number. If she had asked what is the probability that this woman will have 12 males then the method is a good one. But that is not what she is doing. She is looking at the woman after she has already had the 12 sons. There one has to ask what is the probability that this even will occur in the population in general. That is a different question and give a result much close to one. That was my main point. We all have a tendency to focus on what seems to be a rare event and then compute the probably of that rare event. That is backwards and neglects the fact that we have already seen the results before we did the statistical testing/computation.
Ravi and Berry,
I have made a number of simplifying assumptions. The probability of a male child is not 0.5. The experts tell us that the probability of a male child is slightly greater than 0.5. Then when one starts looking at distributions of children by sex for a given woman other factors come into play such as the decision process of the person involved. If parents decide they want at least one boy or at least one girl then the proportion of boy/girl two child families will not be 0.5 as one would expect as many of the families that would have boy/boy or girl/girl will have a third child.
And as you pointed out Berry some women/couples my be predisposed to having boys. There is no reason that probability has to be the same for each person. I’ve glossed over that. Likely the good old central limit theorem comes into play. With the size of the population these things tend to even out remarkably well.
Thanks for clarifying this! (side note: Ravi pointed out the limitation, not me).
My mental rounding capacaties are worse than I thought – the 96 put me way off.
In R, I would calculate:
1/dbinom(x=12, size=12, prob=0.5)
and get 4096. So the odds are more like one in FIVE thousand, not four.
What’s wrong with that?
That’s the probability of having 12 successes in 12 trials, when the probability of success in each trial is 50%. Isn’t that exactly, what the reporter is looking for?
So if every country in the world had 10’000 couples with 12 kids, we would expect the average number of couples per country with 12 boys to be 2.
And by the way: as the binomial distribution is discrete, 12 heads IS a unique result (in statistical sense).
“If the subject of this story come to her attention when she had ten sons she could have written a similar story”.
Yes, but only with odds 1 in 1024. 5 times more likely, in the long run.