I’m pregnant with our 7th baby. So far, our first child is a girl, and the following five children are boys. I used a simple formula I found on Khan academy and figured out that we have 95% (7/128) chance of having one girl and 6 sons, and an 84% (21/128) chance of having 2 girls and 5 sons. Now, I’m trying to figure out what the probability is to have 1 girl first, followed by 6 sons consecutively and in that order. What is the probability we could have one girl, followed by 5 sons, and then another girl and in that order?
7 kids. What is the probability of having one girl and then 6 boys in a row?
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Dr.Peterson
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First, YOUR probability of having GBBBBBB is 50%, since the first six are already determined. That is also your probability of having GBBBBBG. (This doesn't take into account any statistical or physiological conclusions that might be made based on the evidence of the first six births, but only the simplifying assumption that boys and girls are equally likely.)
But the a priori probability (that is, the probability as determined before any children are born, given that you would have 7) of any particular outcome, is 1/2^7 = 1/128 = 0.78%. That includes GBBBBBB and GBBBBBG (again, a priori).
The a priori probability of one girl and 6 boys, in any order, is 7 times that, or 5.47%, because it includes 7 of the 128 possible outcomes. Your 95% seems to be the probability of NOT having that event; perhaps you can show us the formula you can use, and what they say it calculates.
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Is this a real life problem or class-assignment? In real life the chance of having consecutive same-sex children is statistically higher than 50% due to "environmental" factors.
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As Subhotosh Khan mentioned, the basic probability of a pregnant woman having a boy is not strictly 50%. But in the absence of a great deal of other information, lets go with 50% as a good approximation.
If then we ask in advance of ANY birth, what is the probability of the specific sequence GBBBBBB, it is 1/128, which is less than 1%. But that is not impossible, merely quite rare, among those woman who bear seven children.
If we ask in advance of ANY birth what is the probability of a mix of one girl and six boys without regard to birth order, that is indeed 7/128, which means that it is true for only about one woman in twenty of those who have seven children. (By the way, the probability of that NOT occurring is [imath]1 - \frac{7}{128}[/imath], which is indeed about 95%.)
However, you are not asking in advance of any birth. Six births have already happened. Those are no longer matters of probability; they are certainties. A mathematician can only tell you that if births are strictly independent, then your probability of another boy is 50%. Like Subhotosh Khan I suspect that the sex of successive children is not strictly independent for a specific woman, which would mean that the probability of your having another boy is somewhat greater than 50%.
If then we ask in advance of ANY birth, what is the probability of the specific sequence GBBBBBB, it is 1/128, which is less than 1%. But that is not impossible, merely quite rare, among those woman who bear seven children.
If we ask in advance of ANY birth what is the probability of a mix of one girl and six boys without regard to birth order, that is indeed 7/128, which means that it is true for only about one woman in twenty of those who have seven children. (By the way, the probability of that NOT occurring is [imath]1 - \frac{7}{128}[/imath], which is indeed about 95%.)
However, you are not asking in advance of any birth. Six births have already happened. Those are no longer matters of probability; they are certainties. A mathematician can only tell you that if births are strictly independent, then your probability of another boy is 50%. Like Subhotosh Khan I suspect that the sex of successive children is not strictly independent for a specific woman, which would mean that the probability of your having another boy is somewhat greater than 50%.
Dr.Peterson
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My thought, purely theoretical, is that it is conceivable not that previous births influence successive births, but that a series of male births could provide statistical evidence that there is something about either the father or the mother that could change the balance (e.g. sperm formation or the fluids they pass through). That could be genetic or medical or environmental.
I don't know of any data showing the conditional probability of males given all previous births being male, which could be interesting.
In addition, I understand that in reality something like 51%, not 50%, of all births are boys.
Steven G
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According to my wife and my own biology courses, woman only have x chromosomes while the male sperm has both x and y chromosomes. Now if a male's sperm has a y chromosome, then the woman would have a male baby and if the male's sperm has an x chromosome the woman would have a female baby. That is, the woman has no influence on the sex of a newborn.
First, you are of course correct that genetics is not the only cause operating, and that live births among humans favor males by a slight amount.
Dr.Peterson
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This is not, of course, the main issue; and my comments have intentionally been open to all sorts of possibilities. But from what I've read here and there, both mother and father may have some (small) influence. See here, for example, or here. (Just a couple results from a quick search.) I doubt that any such factors often result in a very strong bias, though, unless there is a major biological problem.
Again, my ideas are mostly just hypothetical. The important point is that there can be unexpected causes for bias.
Steven G
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I am a bit confused here. If I toss a coin an awful lot of times, say 8 billion times, no one here would expect that the number heads would equal the number of tails. Why with births are you saying that since there is a 51-49 ratio that there is a slightly greater chance of getting a male?
Dr.Peterson
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Are you saying you don't believe statistics? Or the law of large numbers? The latter doesn't say the observed number for an unbiased coin would be exactly half, but it would be very close!
If you had a sample of 8 billion births, and found that 4.08 billion of them were boys, what confidence interval would you find for the true probability of a boy?
As for the claim (which I'm only reporting third-hand) that around 51% of births are boys, here are more references (which also support the idea of some level of maternal influence): NPR1; NPR2; SciAm; OWD. These indicate slightly different reasons, but they're interesting, and they definitely agree on the fact.
Again, this has absolutely nothing to do with the OP.