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Three children in same family born on the same day

On 26 Feb 2008 the Sun published the following
"Proud Martin and Kim MacKriell never forget their kinds' birthday - because all three were born on the SAME date ... January 29 ..Experts calculate the odds of a couple having three children all on the same date are 7.5 in a million."

The story also appeared in many other newspapers, including the Daily Mail and Mirror. Here I will explain what this statement means and why, and then show that, in fact, there is nothing newsworthy at all in the story. 

First of all, the statement in the article is ambiguous. There are two different scenarios it could refer to:
  1. In a family with exactly three children all three have the same birthday
  2. Three children from the same family have the same birthday

Each of these scenarios has a different probability as follows:
  1. In a family with exactly three children the probability they each have the same birthday is approximately 1/133225. This is indeed approximately equal to 7.5 in a million as stated (although curiously when I asked a number of people to tell me what they understood by the statement "the odds are  ... 7.5 in a million" most people thought it meant 7.5 million to one, to which is very different).
The explanation is quite straightforward. If we assume all three birthdates are independent then the probability that the second child has the same birthday as the first is 1/365. That's because whatever that birthday happens to be (29 Jan in this case) that day is just as likely to be the birthday of the second child as any of the other 364 days of the year.

Similarly the probability that the third child has this birthday is also 1/365. So the probability that all three have this birthday is 1/365 times 1/365 which is equal to 1/133225.

In practice the probability will be higher because parents are more likely to conceive at certain times of the year and so the probability that the second (child's birthday is the same as the first is greater than 1/365). As an extreme example imagine a couple who only make love between May and September. Then any of their children will almost certainly have birthdays between February and June and so the probability of the second child's birthday being the same as the first is more like 1/151.
  1. In a family of more than three children the probability of exactly three having the same birthday is much higher. For example if there are four children (a,b,c, and d), then we can consider four different combinations of three children (a,b,c), (a,b,d), (a,c,d) and (b,c,d). For each of these four combinations the probability of all three having the same birthday is 1/133225. So the probability that at least one of the combinations has the same birthday is 4/133225, i.e. it is four times more likely. With five children there are TEN different combinations of three children so the probability is ten times greater etc.

The final question we need to ask is: is this story newsworthy? The answer is no. For every million families involving at least three children we would EXPECT there to be at least 8 families in which three children share the same birthday.  In the UK there are certainly more than a million families involving at least three children. It would therefore have been far more newsworthy if it was found that NO family in the UK contained three children with the same birthday.



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Norman Fenton


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