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Probability and Combinations

Determine the probabilities of events when arrangement order does not matter

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The Birthday Paradox

Do you personally know anyone who has the same birthday as you? Have you ever seen two people—in a class that consists of just a few dozen people—sharing the same birthday? Given that there are 366 different possibilities for birthdays, a match within such a small- to medium-sized group of people may seem odd. Actually, however, it’s not surprising at all.

Not as Rare as You Think

Go ahead and ask some people around you (not seasoned mathematicians) how likely it is to find a birthday match in a group of 25 people (i.e. at least one instance of two people sharing the same birthday—just the day and month, not year). Most of them will probably tell you that the chance is very small.

However, there’s more than a 50% chance of this happening! To be precise, you just need a group of 23 people for there to be a 50% chance that there are at least two people within the group who share the same birthday. A group of 23 seems so counterintuitively small to most people, but that’s what the math will tell you! If the group size is increased to 30, the probability of a birthday match becomes 70%. For 50 people, it gets very close to 100%.

See the math for yourself: http://www.youtube.com/watch?v=16zXXmSX2v0

Explore More

Let’s now deal with smaller numbers and consider matches in Zodiac signs. Assume that the likelihood that someone has a specific Zodiac sign is simply 1 out of 12. For a group of five people, find the probabilities of the following events:

  1. At least two people have the same sign.
  2. Exactly two people have the same sign, while the other three have three different signs.

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