During my wife's recent pregnancy, it so happened that my wife's 29th birthday fell on the exact same day that her unborn child was 29 weeks old (i.e. it was 29 x 7 days from the date of conception as advised by the doctor)

I would like to know what the probability is of the above event occurring for a randomly chosen pregnant woman – i.e. that the pregnant mum's x'th birthday falls on the same day that the unborn child is x weeks old EXACTLY.

Please assume:

1. The randomly chosen woman is pregnant
2. A doctor can always tell us the exact day on which the child was conceived
3. Women always carry successfully to full term exactly ( i.e. 40 weeks to the day)
4. Any reasonable assumptions regarding the age distribution at which women fall pregnant
5. Plus anything else that I have forgotten that is relevant!

Look forward to hearing the answer

kind regards

Tom

 
Hi Tom.

Let us start by assigning variable a to be the probability the
pregnant women is under 40 years of age, since the number of weeks of
pregnancy is only 40.

Our expression for the probability is "a".

Now assuming that there is no calendar variation in birth rates (any
day of the year has roughly the same number of birthdays as any other
number - which may fail 9 months from Valentine's day), then there is
a 40 week / 52 week chance that the women's birthday falls during the
pregnancy.

Our probability expression is now (40/52)a.

Next, the woman's birthday must fall on the same day of week as the
conception. That's one in seven:

Our probability expression is now (40/52)(1/7)a.

Interpreting things so far, we know the woman is turning an age under
40 during the pregnancy and her birthday is on some whole number of
weeks from conception. Since there are 40 possible "whole number of
weeks" and she has a birthday on just one of those, we apply a 1/40
factor:

Our probability is now (40/52)(1/7)(1/40)a.

That should give us the likelihood you are looking for. If we make
the estimate that 95% of pregnancies are to mothers under age 40, then
a=.95 and so our calculation would make the probability:

P = (40/52)(1/7)(1/40)(.95) = 0.0026 (with rounding)

or about a 1 in 400 chance.

Stephen La Rocque.>