Taking A Look At The Birthday Paradox
The birthday coincidence or paradox results when two people within a group share the same date of birth. Surprisingly, in order to have a fifty/fifty probability of sharing the same birthday, there is only a small group of people required, rather than the more expected thought of needing a large group.
To show how to calculate the probability of a group finding such a match, before setting off to find the perfect birthday present , it is simpler to first find the probability of all the birthdays being different. Consider a group of two people. The first person could have been born on any of the 365 days of the year, while the second must have been born on one of the other 364 days in order to not match. The first person has a probability of 365/365, which equals 1.0, and the second has a probability of 364/365 which is 0.9973. Multiplying these probabilities together gives a net probability of 0.9973 for having different birthdays. Subtracting this number from 1.0 gives a 0.0027 probability of having the same date of birth.
When you add a third person to the group they have only 363 days to miss having the same birthdays as the first two people. Meaning they have a probability of 363/365, or 0.9945 and when you multiply this to the previous results it results in a 0.9918 chance of them having the same birthday. The probability results of 0.0164 (1-0.9836) of having matching birthdays is the result when you add a fourth person to the group.
As you add more and more people to the group, the number of non-matching days available continues to decrease; and the probability of everyone celebrating a different birth day decreases. The probability of there being two children born on the same day increases and continues to multiply more and more, smaller and smaller fractions, a point near 0.5 is exceeded as soon as the group consists of 23 people. And, once you have 80 people in the group, only one time in ten thousand will everyone celebrate different birthdays.
Although it can be interesting to take a topic and dissect it in this manner, most people do not look at birthdays in this type of analytical terms. Instead, it is considered to be a very special day for a very special person, or in the case of multiple births, very special people. A birthday celebration complete with a birthday cake, birthday gifts and perhaps birthday flowers is the most traditional way that people celebrate someone’s birthday; leaving all of the probabilities, percentages and equations to those who prefer statistics over the more sentimental reasoning.
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