# Generalized birthday problem plus spouse

Suppose a group of persons, say of size N (who are all married), are in a room. For any given N, I would like to know the probability of at least two people in the room sharing the same birth date (day/month/year) AND their spouse sharing the same birth date as another person's spouse. Assume that there is an equal chance of a person and their spouse being born on any day of the year and any year between 1930 and 2000 (a 71-year span, or nearly 26,000 possible birthdays, all equally likely).

Thanks. :)

Edit: Sorry, my English was off. I wanted: "...the probability of at least two people in the room sharing the same birth date (day/month/year) AND their spouses both sharing the same birth date (which may or not be the same as their own spouses birth date)." See comments for further clarification.

• Clarification question: If person A shares the same birthday (all of yyyy-mm-dd) with person B, does person A's spouse need to share the same birthday with person B's spouse to count? If not (i.e. it is allowed for person A's spouse to share a birthday with any person in the room), are there any restrictions in the condition on person B's spouse? Jan 8 at 11:34
• Yes, to your clarification question. For example, if person A shares a birth date with person B, and person A's spouse shares the same birth date with person C's spouse, then this does not count. Jan 10 at 7:34
• No. I have some data on people who completed an online survey. I want to detect duplicate records based on their date of birth and their spouse's date of birth. I found a handful of duplicates based on these two variables and want to know how likely that is. I can calculate the probability for just the person but extending this to their spouse seems complicated to me. I wonder if I just square the result? Jan 10 at 12:32
• The year the spouse is born has to be correlated with the year the person was born. If a person is born in 1940, it is not reasonable to assume the spouse is born in any year between 1930 and 2000 with all equally likely. Would it make more sense to just focus on the month and days without including year at all? If you really want to include year, you should make some reasonable assumption about the conditional distribution of year of spouse's birth given year of person's birth. Jan 10 at 14:03
• I wonder how difficult it would be to simulate this with some kind of “if a==b AND c=d” line to denote the double match of birthdays.
– Dave
Jun 24 at 2:39

The probability is not very easy to find. But, I can find the expected number of couples that have matching birthdays.

There are $$M=25933$$ days between 1/1/1930 and 12/31/2000. We can just assume that the numbers $$1$$ through $$M$$ denote these days. So, if I say my birthday is the number 2, then it is understood my birthday is 1/2/1930, etc.
Assume there are $$N$$ people, all are married couples. Therefore, there are $$N/2$$ couples. Order the couples arbitrarily from $$1$$ to $$N/2$$.
Within a married couple, order the birthdays arbitrarily to make ordered pairs. So, the birthdays can be written as
$$(x_1,y_1), (x_2,y_2), ..., (x_{N/2},y_{N/2})$$
where all the $$x_i$$ and $$y_i$$ are numbers between $$1$$ and $$M$$.

The event the second couple has matching birthdays with the first couple can happen in three ways:
$$x_1=y_1=x_2=y_2$$
$$x_1 \ne y_1 \& x_1=x_2 \& y_1=y_2$$
$$x_1 \ne y_1 \& x_1=y_2 \& y_1=x_2$$
Hence, the probability of matching is the sum of the probabilities of those 3 events:
$$\frac{1}{M^3}+\frac{M-1}{M} \frac{1}{M^2}+\frac{M-1}{M} \frac{1}{M^2}=\frac{1}{M^3}+2\frac{M-1}{M^3}$$ This is also the expected value of the number of matching couples from any two couples.
There are $${N/2}\choose{2}$$ different ways to form sets of two couples. Therefore, the expected number of matching couples is $${{N/2}\choose{2}} \left( \frac{1}{M^3}+2\frac{M-1}{M^3} \right)$$

If you started with, say, 1 million people (500,000 couples), the expected number of pairs of couples that would have matching birthdays is about 0.06. Since this is a very rare event, you will almost never see 2, 3, 4, etc. pairs of matching couples. You can assume it will almost always be either 0 or 1 pair matching. If you repeated this over and over again, you would see 1 pair of couples about 6% of the time and 0 couples 94% of the time.

If you want to consider months and days only, I would ignore February 29 and take $$M=365$$. With $$N=1000$$, you would find the expected number of pairs is about 1.87. So, now you might be able to see 2, 3, 4 etc. such pairs of matching birthday couples. So, you can't use this approach to approximate the probability of seeing 1 or at least 1 pair of couples with matching birthdays.

• How would the answer change if you take into account the dependence between spouses birthyear? Jun 24 at 2:40
• In addition to @kjetil's point, because populations have changed so much over those 70 years, it is essential to account for that in any such calculation.
– whuber
Jun 24 at 13:08