Shared Birthdays

If there are 30 people taking part in a 'Zoom' meeting, what are the chances of at least two participants having the same birthday?

You might estimate that there is about an 8% chance of this happening because there is a 1 in 365 chance of any one person of having a birthday on a particular day. So, in a group of 30 people, the probability would be increased by 30 or 30/365.

The actual probability is 71%! The reason for this is that in our original estimate we were only comparing two people, but in a group of 30, there are actually 435 comparisons or pairings.

Let's see how this works with a smaller group of people (Alex, Bertie, Charlie and Dee) who were all born on a weekday:

If we were to use our original method for estimating the probability that at least two of them were born on the same weekday, then we would come up with an answer of 80% (1 in 5 chance of any two people being born on the same weekday multiplied by 4).

However, the probability is greater than this because everyone should be compared to everyone else. With a group of four people, there are six possible pairings:

Table 1: Pairings for four people

While it is possible to use this method to work out the number of pairings for any group size, it does become a bit arduous when dealing with large numbers. Luckily, there is a formula you can use for calculating the number of pairings. The number of possible pairings in a group of n people is:

n(n-1)/2.

The pairings don't have to be in the order shown in Table 1. They can be in any order as long as they include all the pairings. Table 2 shows the six pairings in a different order and a slightly different format:

Table 2: A different order and format

So, what is the actual probability of at least two members of this group being born on the same weekday? A tree diagram is one way of working out the probability (see Figure 1).

Figure 1: A Tree Diagram

In this tree diagram, Bernie's birthday is compared with Alex's. There is a 1 in 5 chance of them being the same and a 4 in 5 chance of them being different. The 'Yes' branch doesn't continue because the problem is to determine the probability of at least two people having the same birthday. So, if probability takes us down a 'Yes' branch, there's no need to go any further.

Charlie's birthday is then compared with Bertie's and Alex's. Then, Dee's birthday is compared Charlie's, Bertie's and Alex's.

Tree diagrams can get a bit complicated, so I am going to look at the same question using tables. I found that using tables helped me understand tree diagrams better.

There are a few things we need to bear in mind before we start:

• If one person is comparing with a number of other people, then the matching probabilities are added together.
• For the same comparison, the 'match' probability and the 'no-match' probability add up to one.
• Each set of comparisons ('match' or 'no-match') are multiplied together to get the overall probability.
• The 'no-match' branches of the tree diagram are less complicated and, therefore, easier to calculate than the 'match' branches.

Each pairing has a 1 in 5 chance of both people being born on the same weekday as shown in Table 3.

Table 3: Pairings and the chances of a match

If we add the probabilities together, we find that there is:

• 1 in 5 chance of Bertie having the same birthday as Alex,
• 2 in 5 chance of Charlie having the same birthday as Bertie or Alex,
• 3 in 5 chance of Dee having the same birthday as Charlie, Bertie or Alex.

Table 4: Addition of probabilities

We can calculate the 'no-match' probabilities of by subtracting the 'match' probabilities from one as shown in Table 5.

Table 5: No-match probabilities

Notice, both in the tree diagram and Table 5, that for n people the 'no-match' probabilities start at n-1/n and that the dividend (the number being divided) is reduced by 1 for every subsequent set of comparisons.

We get the overall 'no-match' probability by multiplying 4/5 x 3/5 x 2/5. This gives an answer of 24/125 or 19%. The overall 'match' probability is 81% (100% - 19%).

Using tree diagrams or tables can get quite difficult for larger numbers like 30 people and 365 days. There are, of course, mathematical formulae for solving this problem — see the Birthday Problem on Wikepedia.

Computer programs can also find the answers by modelling the processes described above. I have written a Shared Events web app which you can use to solve the Birthday Problems and many other variations.

The following table summarises the data for groups of 2-30 people:

Table 6: Data for 2-30 people

These methods don't take leap years into account or the fact that birthdays are not spread evenly throughout the year.

Written with the help of an explanation by Elouise Wills

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