# Two women, same due date, giving birth on same day?

I'm trying to figure out the chance that two women with the same due date, give birth on the same day.

Do I just multiple the chance of giving birth on any given day for each woman, for each day, and add them together. For example, if the chance of giving birth at 39weeks 6days is 4%, 40 weeks is 5%, 40 weeks and 1 day is 4%, etc. The calculation would be: (...+ 0.04*0.04 + 0.05*0.05 + 0.04*0.04 +...)?

And if I wanted to know the chance for 3 women, I would do the same for woman1&woman2 + woman2&woman3 + woman1&woman3, and then add all 3 sums together?

• It looks like you are willing to make simplifying assumptions, such as that the births are (statistically) independent. As a probability exercise that's fine. But if you're using this calculation to derive information about actual births, it seems likely it could produce deceptive results because there are some well-known causes of strong interdependence among birth dates: doctors tend not to want to deliver babies on weekends; women living or working closely together often have cycles in phase; etc. These seem mostly to work to increase the chance of births on the same day. – whuber Mar 31 '15 at 23:01
• One important consideration is that for the probability calculation to be meaningful at all, the event (they give birth on the same day) would need to have been specified in advance. If it's after the fact, such calculations (which are performed as if the event were specified first) can produce impressive but meaningless numbers. – Glen_b Apr 1 '15 at 1:46
• When you say "the chance for three women" what event do you mean? All three give birth on the same day? At least two out of three give birth on the same day? Something else? – Glen_b Apr 1 '15 at 1:49
• @whuber FYI (as a total side note), my understanding is that the claim that women who live together often synchronize cycles appears to be wrong. – djs Apr 1 '15 at 8:54
• @Dougal Thank you for the correction and the link to an interesting site. Note that whether one particular speculated cause of dependence is true or not does not affect the possibility of other causes of dependence, so the need for caution remains. – whuber Apr 1 '15 at 15:42

Under an assumption of independence of the occurrence of the two birth events, you're correct, you just square each of the probabilities of birth on each given day and add them across the possible days $\sum_i p_i^2$
This question is unclear. If you mean "with three women what's the probability of exactly two giving birth on the same day?" then that would be the $\sum_i 3p_i^2(1-p_i)$. If you mean "what it the probability of at least two giving birth on the same day?" it would be the previous one plus the probability of all three (i.e. add $\sum_i p_i^3$).
Note that this is not quite the same as what you suggested ($\sum_i 3p_i^2$) because that counts the "all three give birth on the same day" event three times.