# Testing the Birthday Paradox

The birthday paradox states that in a room of just 23 people, there is a 50/50 chance that two people will have same birthday. In a room of 75, there is a 99.9% chance of finding two people with the same birthday. In this experiment, you will evaluate the mathematics behind the birthday paradox and determine whether it holds true in a real world situation.

### Problem:

This experiment explores the mathematics behind the birthday paradox and determines whether it holds up in the real world.

### Materials:

- Approximately 150 test subjects
- Calculator
- 150 notecards
- Pen
- Notebook for recording and analyzing results

### Procedure

- To understand the birthday paradox, begin by evaluating a simplified version of the problem:
- What is the probability that within a group of three, two or more people were born on the same day of the year?
- In many cases, probability problems can be solved by analyzing the complementary problem. In this case, the defined problem is the probability that two of the three people will have an identical birthday, and the complementary problem is the probability that zero of the three have the same birthday.
- Each one of the three people can have a birthday on any one of the 365 days of the year. Thus, the total possibilities can be calculated by multiplying (365x365x365).
- To determine the probability that two of the group members have a common birthday, first evaluate the probability that two people in the group do not share the same birthday. This value should then be subtracted from one.
- Figure out the number of ways that no two people have the same birthday. There are 365 possible days for the first person’s birthday. There are then 364 possible days in the year that allow the second person’s birthday to be different from the first person’s birthday. Likewise, the third person’s birthday must be one of the other 363 days in the year in order to not share a birthday with the other two in the group.
- Therefore, the probability that there will not be an identical birthday in the group of three: (365x364x363)/(365x365x365), or 0.992.
- It appears to be very likely that no two members of this group will have the same birthday, because the probability of two or more members having a common birthday is (1 - 0.992), or 0.008. This means that there is approximately a 1 in 125 chance that at least two people in the group of three will have the same birthday.

- Using the same mathematical analysis, calculate the probability that two people will share the same birthday when you increase the group size to 23 people.
- What happens in your mathematical analysis when you increase the group size to 75 people?
- Test the validity of the birthday paradox in the real world.
- Record the birthdays of 150 people on note cards.
- Shuffle the notecards and randomly select 23 cards.
- How many of the people in this “group” share a birthday?
- Repeat steps 6 and 7 several times. How often do you find a shared birthday among the 23 people? Does it appear to be 50/50?
- Now, shuffle the notecards and randomly select 75 cards.
- How many of these people share a birthday?
- Repeat steps 9 and 10 several times. Do you find that at least two of the 75 people share a birthday 99.9% of the time?

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