Don't calculate, simulate - the birthday problem

Although we have extensively demonstrated in past blogs the advantages of simulation over arithmetic computing, there are areas where even mathematical artists reach their limits and better resort to simulations. A prime example is the arithmetic with permutations.

Here's the well-known birthday problem: There are 23 unrelated people at a party. What is the likelihood that at least two of these guests will celebrate their birthday on the same day? (We assume that one year has 356 days and none of the guests are born on a leap year)

A common search via the well-known search engines reveals various solutions, whose numbers have been put aside the famous permutation sign (!). The list of concrete formulas for solving the birthday problem will not been shown here, there are all too complicated for us.

In the following Excel we have listed our 23 guests who can have their birthday between 01.01 (day 1) and 31.12 (day 365). In cell F4 we used a matrix formula of Excel, which determines the number of multiple entries. We present the result of a simulation with 10,000 iterations. So we see that the probability that at least two guests have birthday on the same day is over 50%! Well, would you have expected the result?

Monte Carlo simulation birthday problem Excel MC FLO

Instead of relying on arithmetic calculations using permutations that most people perceive as difficult (so do we), a simple formula in Excel in combination with a simulation is enough to achieve the result.

Well, the birthday problem is emblematic for a variety of problems where the probability is misjudged and which is used in practice to test designs or experiments.

Specifically, quite analogously: Let us imagine that a machine consists of 365 parts and that such machines are located at 23 locations. As part of a quality control, exactly one part of a machine should be checked for each site, with no part being allowed to occur several times. Would you order the site managers to randomly select a part? With the above solution to the birthday problem, you now know that such a procedure is not advisable.

Or better: if you bet on two soccer players having brithday on the same day (given the the starting line-up of both teams, including the referee), you will probably win the bet with a more than 50% chance. Have fun!


Note: We have used MC FLO with german language settings enabled.

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