# The Surprising Number of People Needed in a Room for Two to Share a Birthday

Have you ever encountered someone who shares your birthday? How about in your workplace? The concept of shared birthdays, known as the birthday problem, raises questions about our understanding of number theory, probabilities, and our assumptions about the world. It highlights the counter-intuitive nature of mathematics for many people.

The birthday problem poses the question: “What is the minimum number of people in a room to have a better than 50% chance of two people sharing a birthday?” It seems like a simple question, but the answer is puzzling.

Initially, when math students are introduced to the birthday problem, most of them believe that a group of 183 people is needed to have a better than even chance of two people having the same birthday. Their reasoning is based on the assumption that they only need to compare others to themselves and find a match. However, when they realize that not every combination has to involve themselves, it becomes clear that the number needed is lower than 183.

The combinations do not scale linearly as more people are added. Through an interactive simulation, it becomes evident that only about 23 people are needed for a greater than 50% chance of a shared birthday, despite there being 15 times more days in the year.

Large numbers can be difficult to comprehend. The COVID-19 pandemic highlighted our limited understanding of exponential growth when presented with models of unchecked spread. For example, choosing one cent doubled each day for a month would result in approximately 10 times more money than taking \$1 million on the first day.

The power of exponential growth can also be seen in the story of the supposed inventor of chess, who requested rice from a king. By placing a single grain on the first tile and doubling it each time, the amount of rice on the final tile quickly surpasses the world’s rice supply.

Mathematics often defies intuition. In Monte Carlo in 1913, there was a run of 26 straight black results on roulette wheels. Despite the improbability, gamblers increasingly bet on red, assuming it was “due.” However, each spin of the wheel has no memory of previous outcomes, so the chance of red appearing does not increase over time. Many people lost money that night due to this misconception.

However, some situations do depend on previous probabilities. The Monty Hall problem, where you choose a door to win a car, demonstrates this. After being shown a door with a goat, the odds of winning if you switch or stay are not 50/50. The likelihood of your next choice is influenced by your previous choice.

Being numerate, or having the ability to reason with numbers and apply this reasoning in various contexts, is crucial. Developed numeracy skills have correlations with better life outcomes, including employment, income, health, and well-being. Being numerate allows for a deeper understanding of how the world works, even if it may seem counter-intuitive. It also enables individuals to make informed decisions in different situations.

So, continue to engage with mathematical and numerical problems. They may prove useful one day.