BIRTHDAY QUESTION PROJECT

The notion of likelihood is a major component of our everyday lives. How likely is it that a particular scenario will actually happen? What are the chances? Sometimes the answers to such questions are surprising and counterintuitive. Is it a coincidence to run into an acquaintance at an airport or to find that you share a birthday with another person? Perhaps the likelihood of these kinds of events is higher than we would initially think! Let's consider this birthday question:

How many people are needed in a room so that the probability that there are at least two people whose birthdays are the same day is roughly one-half?

Let's pretend that there are no leap years and assume that it is equally likely to be born on one day as on any other day.

  1. Provide an initial guess! How many people would you guess are needed in a room so that the probability that at least two people share a birthday is about 50%? Explain your thoughts.

First, let's consider a room with only two people.

  1. Using the Counting Principle, how many pairs of birthdays are possible?
  1. How many of these pairs of birthdays have the property that both dates are different?
  1. Determine the probability that two people do not share the same birthday.
  1. Determine the probability that two people have the same birthday.

Consider a room with only three people.

  1. Using the Counting Principle, how many triples of birthdays are possible?
  1. How many of these triples have the property that all three dates are different?
  1. Determine the probability that all three people do not have the same birthday.
  1. Determine the probability that at least two of the three people have the same birthday.

Consider a room with only four people.

  1. Look for patterns! Use the above steps to determine the probability that at least two of the four people have the same birthday. Your answer, correct to five decimal places, should be equal to .

While the probability of having a pair of matched birthdays among four people is still nowhere near one-half (0.5), it is almost twice as large as the probability of finding a birthday match among three people.

  1. Continue to calculate the probabilities and fill in the table below. Instructions for creating an Excel spreadsheet are included at the end of this project.

Number of People in the Room / Probability of At Least Two Sharing the Same Birthday (correct to five decimal places)
2
3
4 / 0.01636
5
10
15
20
25
30
40
50
60
70
80
90

Notice how quickly the probability heads toward 1. When there are only 50 people in the room, it is nearly a sure thing that there will be a pair of people that share a birthday. When there are 90 people in the room, we can be essentially 100% confident of a match; yet 90 people is a far cry from 366 people, which guarantees a match (the pigeonhole principle).

  1. Answer the birthday question. How many people are needed in a room so that the probability that there are at least two people whose birthdays are the same day is roughly one-half?
  1. How does the answer to the birthday question compare with your initial guess?
  1. Are you surprised by the answer to the birthday question? Explain.
  1. Share an example of another event that would feel coincidental if were to happen to you.
  1. Could knowing the probabilities of an event ever help you make decisions? Explain.

Suggested spreadsheet instructions

Let column A contain the number of people in the room. You will need 2 through 90 so that you can fill in the table on exercise #11.

Let column B contain the probabilities that at least two of the people in the room share a birthday. The formula will use the "PERMUT" Excel function to create products like:

etc.

Formula for Column B:

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