I am a twin who celebrated my 13th birthday on Friday the 13th. Is this rare? What is the probability of this occurring?

Question

I was born a twin who celebrated my 13th birthday on Friday the 13th many years ago. Is this rare? What is the probability of this occurring? I am a senior who finally wants to know the answer. Can you help? I realize it is quite common for a single birth, but for twins born 60+ years ago? So much to consider in solving this. I am afraid it is beyond my math skill level. Thanks for your consideration James

Answer 1

There are many factors, and depending on what factors you deem to be important this will affect the answer.

For simplicity's sake, we can probably safely assume that the date of birth is independent of whether or not you were born a twin. Further, we can probably safely assume that each day in the calendar is equally likely to be a randomly selected person's birthday. Finally, we should probably ignore child mortality rates and assume that death rates are independent of date of birth and being a twin.

In a bizarre alternate world where all twins are sacrificed according to some religious ritual by their tenth birthday, the probability that a twin would celebrate their thirteenth birthday on friday the thirteenth would of course be zero. This is an extreme example of when if deathrates are not in fact independent, we could have skewed results. We don't technically know if they are independent, but I see no reason to believe otherwise or that it would otherwise significantly affect the results.

Many sources cite that about 32 people out of every 1000 is a twin.

As discussed in my answer here, if we assume all days in the calendar are equally likely birthdays, chance of having 13th birthday on friday 13th is the same chance as having been born on a friday the 13th. Our calendar repeats and friday the thirteenths do occur at a slightly above average rate at exactly $688$ occurrences out of $146097$ days (the length of time until the calendar repeats again).

Given our assumptions then, the probability that a randomly selected person happens to be a twin who celebrated their thirteenth birthday on friday the thirteenth is approximately

$\frac{32}{1000}\cdot\frac{688}{146097}\approx 0.00015069$, or about one in $6635$ people.

Compare this to the more naive calculation $\frac{32}{1000}\cdot\frac{1}{7}\cdot\frac{12}{365.2425}\approx 0.00015019$ where we incorrectly assume day of the week is independent of day of month. It is ever so slightly less, but still a close approximation. As a further comparison, about one in 212 people have their thirteenth birthdays on a friday the thirteenth.

In reality, the assumption that every calendar date is equally likely is probably not valid, but it allows us to do our calculations. There are generally more births during the fall. The exact amounts though are relatively close, and there is a thirteenth of the month in every month, so the true figures should not vary from the above results by any significant amount.

Answer 2

Well, the probability of your 13th birthday being on a Friday is 1 in 7.

The probability of being born and the 13th is 12 in 365$\frac 14$.

So the probability of one's 13th birthday on Friday the 13th is $\frac 17*\frac {12}{365\frac 14} = \frac {16}{3409}$

The probability of your being a twin is... I have no idea.