Birthday Party Depot

Revealing The Mysteries Of The Birthday Paradox

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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 start adding more people to the group the odds naturally change. For example, a third person added has to miss having the same birthday as the first two and has only 363 days in which to do so and that changes the probability to 0.9945, or 363/365, and multiplying it to your previous results gives you a 0.9918 chance of them having the same birth day. And, of course, it changes again when you add a fourth person to the equation, 1 – 0.9836= 0.0164 which is over 1%.

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.

Even though it may be interesting to learn different facts and discover unusual tidbits about ordinary events or processes, most people would probably prefer to celebrate the uniqueness of certain celebrations such as a birthday, with the exception of multiple births. It is a unique, special event for one special person on one special day. They prefer to celebrate in the more traditional way of throwing a birthday bash, complete with presents, cards, some party favors and a birthday cake. The also prefer to leave all of the analytical aspects, such as equations, percentages and probabilities to those who are less sentimental.