Birthday paradox
The birthday paradox, also known as the birthday problem, is a probability phenomenon which states that in a group of randomly chosen individuals, the probability of two or more of them having the same birthday is actually quite high. The chance is more than 50% when the number of people reaches 23.
The birthday paradox is not a true paradox: It gives an unexpected result — one that goes against our intuition. Its an example of a non-intuitive result which can arise from using statistics to calculate probability.
Though it appears counterintuitive, you might find it to be surprisingly accurate. The idea is that when you have a larger group of people, the chances that two people have the same birthday are quite high.
To explain this concept, lets break it down into two parts: the probability of two people having the same birthday, and the number of people required to reach a certain probability.
When two people are in a room, they have a 1/365 chance (or 0.27%) of having the same birthday. Thats because there are 365 total days in a year, so the chances of any two people picking the same day is 1/365.
Now, if we add a third person to the room, the probability of two of them having the same birthday goes up significantly. At this point we have three possibilities: two of them could have the same birthday, all three could have the same birthday, or none of them could have the same birthday. So the chances of two of them having the same birthday become 3/365 (or 0.82%).
As more people are added to the group, the probability of two people having the same birthday increases exponentially. When you get to 23 people in a room, the probability of two or more of them having the same birthday is 50.73%.
So why does the probability increase so rapidly? Its because the number of possible unique combinations (of birthdays) increases. For example, in a group of three people, there are three possible unique combinations: one person with a different birthday from the other two, two people with the same birthday and different from the third, and three people with different birthdays. In a group of 23 people, the number of possible combinations is 253, so the chances of two or more people having the same birthday is much higher than in a smaller group.
Its interesting to note that the probability of two people having the same birthday never reaches 100%. Its not possible for every combination of birthdays to be unique, but it is possible for every single person to have a different birthday.
The birthday paradox can be used as an example of how statistics can be used to calculate probability and how our intuition can be wrong. Its a counterintuitive result, and yet it turns out to be true. Knowing the birthday paradox can help us understand the power of statistics and avoid making decisions based solely on intuition or conventional wisdom.
