he Birthday Question
The notion of likelihood is a major component of our everyday lives. How likely is it that a certain scenario will actually happen? What are the chances? Sometimes the answers to such everyday questions are surprising and counterintuitive. Is it 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 events happening is higher than we would initially think! Let’s focus on the birthday scenario and work towards answering the following 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.
2. Using the Counting Principle, how many pairs of birthdays are possible?
3. How many of these pairs have the property that both dates are different?
4. Determine the probability that two people do not share the same birthday.
5. Determine the probability that two people have the same birthday. Hint: P(A) = 1 – P(not A).
Consider a room with only three people.
6. Using the Counting Principle, how many triples of birthdays are possible?
7. How many of these triples have the property that all three dates are different?
8. Determine the probability that all three people do not have the same birthday.
9. Determine the probability that at least two of the three people have the same birthday.
Consider a room with only four people.
10. 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, it is almost twice as large as the probability of finding a birthday match among three people. Continue to calculate the probabilities in this manner and fill in the table. Consider using an EXCEL spreadsheet to help with the calculations – save and submit the file along with this worksheet.
|Number of People in the Room||Probability of at Least Two Sharing the Same Birthday (correct to five decimal places)|
It is truly surprising how quickly the probability heads toward 1. With only 50 people, it is almost a sure thing that there will be a match. With 90 people, we are essentially 100% confident of a match; yet 90 is a far cry from 366 people, which guarantees a match for sure.
11. 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?
12. When events happen that feel random or seem to have a low probability of occurrence, we tend to call them coincidences. Describe a coincidence that you have experienced in your life.
13. While we may not be completing probability calculations in our minds from day to day, we still use probability regularly. Any time that we consider the likelihood of various outcomes to help us make decisions, we are using probability! Share an example of how probability is used in your everyday life.