Take two people at random, what is the chance their birthdays are on the same day in 2023? Before you answer 1/365, think again. Scroll down for the answer.
To work out the probability that the two people have the same birthday and that it occurs in 2023, we need to total up all the possible cases where the answer is yes versus all the possible combinations. We need to look at four years, three ordinary ones and one leap year, in order to get the full picture, as one person might have been born on February 29th.
If the first person was born on January 1st then there are 4 cases where the second person was also born on that day in one of the four years. The same is true for every day of all four years, except for February 29th, which does not occur in 2023 and hence gives no matches. That gives us 4 x 365 cases each year over four years. So the total of the yes cases is 4 x 365 x 4 = 5,840.
How many total combinations are there? The first person could have been born on any of 365 x 4 + 1 days = 1,461, as that is the number of days in four consecutive years. To get the total possibilities we need to square this number, as the second person could also have been born on any of these days. So the total combinations amount to 1,461 x 1,461 = 2,134,521.
Our answer is 5,840/2,134,521 = 0.002736. This is slightly less than 1/365 = 0.002740, as the February 29th possibility decreases the chances of a birthday coinciding in 2023. As expected, it is slightly more than 1/366 = 0.002732.
Probability is tricky. The human brain is not well suited to calculating it correctly. Our intuitive answers are often wrong.
NB Even the above calculation needs to be adjusted, because not every 4th year is a leap year. Years divisible by 100, but not divisible by 400 are not leap years. Thus 2000 was a leap year but 1900 was not.