Probability of the Same Date of Birth

Todennäköisyys samasta syntymäpäivästä

Human physiology offers a familiar example of how poorly we intuitively grasp probability. People routinely lose money in games of chance and betting, largely because the human mind struggles to distinguish probability from proportionality.

Consider a scenario from the world of football. Two teams line up against each other, with 11 players per side, giving a total of 22 players on the pitch. The question is:

What is the probability that at least two of these players share the same birthday?

This is a classic instance of the birthday paradox, and the result is surprisingly counter‑intuitive: with 22 individuals, the probability of a shared birthday is already about 52 per cent — meaning it is more likely than not that two players were born on the same day of the year.

The probability that two individuals share the same birthday is 50.7 per cent. This arises because probability theory does not examine individuals in isolation, but rather the pairs that can be formed between them. With 23 people, there are 253 distinct pairs. The first player can be paired with any of the remaining 22, the next with 21, and so on. Although the likelihood of a match within any single pair is small, the sheer number of possible pairings pushes the overall probability above fifty per cent.

The expected gain from such a wager is modest, yet the chance of winning is still better than that of the opposing side. The ratio between 253 possible pairs and 365 days offers an intuitive sense of why the result is so counter‑intuitive. When decision‑making is guided by mental imagery rather than actual calculation, one becomes easily led astray. In a real situation, many people would likely accept the bet, because intuition fails to account for the combinatorial growth in pairings. This illustrates how difficult it is to distinguish reality from the mental picture we construct.

Calculation There are 11 players on each football team. If we assume one substitute per team, the total number of players becomes 24. With 24 individuals, the probability that at least two share the same birthday rises to 54.1 per cent.

364/365 = 0,997

363/365 = 0,995

" - "

342/365 = 0,937

By multiplying the 24 successive fractional terms that represent the probability of all birthdays being different, we obtain the complementary probability of a match. The result is 27/50, meaning that the chance of at least two individuals sharing a birthday is 54 per cent. In other words, the odds of winning such a wager are modest, yet still slightly in one’s favour.

If 60 players were to gather on the pitch for an awards ceremony, the probability would be virtually 1: in a group of that size, it is almost certain that at least two people were born on the same day. This follows from the explosive growth in the number of possible pairings as the group size increases.

The same phenomenon appears in smaller groups as well. The probability that two people share a birthday in a room of 30 individuals is already 71 per cent. This is a classic illustration of the birthday paradox: human intuition severely underestimates how rapidly the number of potential pairs grows relative to the size of the group.

31.5.2026*17:40 (1075 - 121)
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