## 12 comments on “The Birthday Problem”

1. That’s really interesting to see – I’d like to see the distribution over years as well. Like before inductions became as common, as well as more advanced birth control. With Will I was “encouraged” to consider induction so he wouldn’t be born while the doctors were on their Thanksgiving holidays, but he came on his own a week before.

2. Using the frequency data referenced here (http://www.panix.com/~murphy/bday.html), I found no significant difference from the theoretical value (assuming uniformity) for P(23) = 0.507. I just presented this as a teaser on Mon night to kick off the summer term of an MBA course; I’ll hit them up with this “update” in an hour.

3. It is an undergraduate exercise to show that if one date has probability 1/365 + delta and another 1/365 – delta, with all other dates having probabilities 1/365, then the probability that there is one match in a group of n is greater than if all probabilities are equal. By extension, if the probabilities are unequal, then the probability of a match in a group of n is greater than if all probabilities are equal. The extreme case is of course, everyone being born on the same date!

4. Ran a quick and dirty monte carlo simulation in Matlab. Here’s what I got.

```P( 2) = 0.002725000000000

P( 5) = 0.027205000000000

P(10) = 0.117384000000000

P(20) = 0.412472000000000

P(30) = 0.707320000000000

P(40) = 0.891747000000000

P(50) = 0.970613000000000

P(60) = 0.994158000000000
```

here’s the code I ran. Not the best but it’ll do:

```hit = zeros(8,1);

idx = 0;
for k = [2 5 10 20 30 40 50 60]
idx = idx + 1;
for i = 1:1e6
[~,x] = histc(rand(1,k),[0;cumsum(p(:))/sum(p)]);
if ( length(unique(x)) == length(x) )
hit(idx) = hit(idx) + 1;
end
end
end

1 - hit/1e6
```
5. Could babies be induced to avoid being born in the “wrong” Chinese year. I am told some animals are good to be born under some like the pig bad.

Is there a bump caused by Valentines day? Would you expect the superbowl to cause an increase or a drop?

There is a chapter in Wiseman’s book quirkology where he talks about the parents of Churchmen faking their birthday to be on December 25th

6. Regarding your conjecture about tax breaks, I’d be more inclined to suspect it’s a backlog of planned C-sections due to doctors taking off the preceding few days (very low frequency compared to neighboring points, no doubt a holiday “seasonal” effect).

7. “The probabilities depend on who is in the room” One other issue is some groups are more likely to share birthdays than average. Professional sports people tend to be born at a time that means they are old for their underage games. So they are on the old end of 10 in the under 11′s. This means they are usually born in the first three months of the year.

Kary Mullis in his support for astrology says “A recent scientific study of the distribution of medical students in birth
months discovered that a lot of medical students were born in late June. ” http://www.crawfordperspectives.com/documents/IAMACAPRICORN_000.pdf

I dont know the paper but it implies that there could be many professions that cluster in birthdate

8. I coincidentally did a simulation the other day. The answer is essentially unchanged, but for medium groups (10-50 people) reality seems to be very slightly favored, to the tune of 0.15% more likely to find a match. My simulation: The CDC data includes birth day for 1969 to 1988.

I actually did a very similar simulation a few days ago using that full data and found that they were nearly identical. Slightly more likely in reality than the simulation, but only 0.14% more likely at n=23 (and n=23 is still the minimum group size necessary for >= 50%.