Probablity Question

A question on calculating some odds:

The chance that the last four digits of one’s Social Security number will match their child’s birthday. For example, SS# xxx-xx-8306, with their child being born August 3, 2006. Assume that there was no planning of the expected due date involved. Any way this can be reasonably figured out?

First, it needs to be a birthday expressible in four digits. There are 9x9, i.e., 81 such days out of the 365 days of the year (ignoring leap years). Second, the last four digits need to match. Given a particular birthday, that’s a 1/10,000 chance. So the probability is 81/3,650,000, i.e., about 1/45,000.

Sure, if you assume that your child has a single digit month, single digit day birthday, e.g. 8/3/06 and the last four digits of your SSN are randomly generated, the odds are just 1 in 10,000. No different than the odds of guessing any four digit number in one try.

Now, if you ask what the odds are of finding a person in a given population with a birthday that matches their SSN, that gets a little trickier, as some people will have five or six digit birthdays, but just off the top of my head I’m guessing it will just be a prefactor on the 1/10000 percentage based on the fraction of four digit birthdays to the others.

Edit: ::shakes fist at Giles::

And with multiple parents and multiple kids, the odds get even better!

Using your numbers, there’s about a 1/1200 chance that either Jim Bob or Michelle Duggar has a SSN that meets those criteria.

One thing to consider is how many ways 4 digits can be interpreted. The more ways, the more likely for a match.

Example: 0512 could be May 12 or 5 December. If you want to increase the chances, allow for backwards and random combinations (that’s what makes the Bible Codes more likely).

And what happens to numbers like 9998? Are they discarded before odds calculations? Or rearranged? Added?

So we have to know your rules before calculating the odds. Or evens, even.

Wow, beaten to the narrow answer already.

BUT, getting on to the broader answer, you need to realize that 81/3,650,000 is the chances, specified ahead of time, of that particular match.

If you’re just noticing a particular coincidence after the fact, remember, there is also a chance the birthday would match with European numbering (day-month-year), or year-month-day, and a chance the four SSN digits are exactly the same as the four-digit year, and the chances of all the other ways a birthday could be somehow matched to four digits. So if you’re asking ‘what are the chances a birthday could match the last four digits of the person’s SSN?’ then the answer is really “Well, always, if you’re creative enough to find a particular way of matching them.”

Are all four digit combinations equally likely in real SSNs? If not, the exact distribution is essential information.

Ok, thanks much for the responses so far. For our purposes here, let’s assume it’s a four-digit birthday, and we’re talking about the M/D/YY numbering convention.

Yes, this is a straight serial number, assigned from 0001…9999 (so all zeroes is not used).

The area number (first 3 digits) and the group number are assigned to particular locations and in certain orders, so all combinations are NOT equally likely. In fact, from those first 2 groups, you can determine the likely location & approximate birth year of a person. Or you can determine that it’s a fake SSN; many combinations have never been issued (yet).

I assume that 9998 would just be September 9, 1998, or am I missing something?

Oh, so Giles’ otherwise excellent post very slightly underestimated the odds. It’s actually 81/365 x 1/9,999. Oh heck, while we’re at it we can also account for leap days (since the day would be 29, it couldn’t match so the leap day adds to the denominator but not the numberator), so it just : It’s [814/(3654+1)] * 1/9,999. Which of course is still Giles’ 1/45,000 to any reasonable precision.

Since when does “99” equal “September”? And if it does, what do “0998” and "9098"mean?

If they mean the same to you, you have just changed the odds calculation.

Ref the OP’s post #8, the digits are to be parsed as mdyy. In that format, “9998” is equivalent to month=9, day=9, year=98, so equals September 9 1998.

Musicat has stumbled upon a point, though, that some SS#s can’t be interpreted as dates - particularly those with zeroes in the month and day slot. Though it looks to me that that wouldn’t affect these calculations at all.

I think it would.
Using a mdyy format, Sep 9 is the highest possible day. That excludes about 111 of the 365 days of the year, which is significant.

Looks like your year only has 9 possible months. Yippee! No mortgage payments after September!

Why not? If some numbers are excluded from either numerator or denominator of an odds calculation, that affects the resulting math, doesn’t it?

In addition to September 9 being the highest possible day, you also have days 10 and up for every month that are not represented. That’s how Giles came up with only 81 possible birthdays that could be represented in that format. His calculations were mostly correct, but should be adjusted by the fact that “0000” is not a possible SSN, and that there are 366 days in some years. Quercus correctly figured it.