The Probability of Finding Two People Born in the Same Month in a Group

In this article, we will delve into the fascinating problem of calculating the probability of finding at least two people who were born in the same month in a group of ten people. We will explore different methods to estimate this probability and provide a detailed analysis for a group of six people.

Basic Probability Estimation

Let's start by considering the group of 10 people. The problem can be approached by calculating the probability that each person was born in a different month. This will help us find the complement, which is the probability that at least two people share the same birth month.

This can be calculated as follows:

First, we calculate the probability that each person has a unique birth month. With 10 people and 12 months in a year, the probability that the first person is born in any month is 12/12, the second person in any of the remaining 11 months is 11/12, and so on. Therefore, the probability that all 10 people have different birth months is:

12/12 × 11/12 × 10/12 × 9/12 × 8/12 × 7/12 × 6/12 × 5/12 × 4/12 × 3/12 ≈ 0.00906.

The complement of this probability, which is the probability that at least two people share the same birth month, is:

1 - 0.00906 ≈ 0.99094.

This is a rough estimate, as it assumes that the number of days in each month is the same, which is not precisely true. However, it provides a good approximation.

Using Combinatorial Methods

Another method to solve this problem is to use combinatorial methods. For example, calculating the number of ways to choose 2 people out of 10 to share the same birth month can provide an exact probability. We know that there are 45 ways to choose 2 people out of 10:

10C2 45.

The exact probability can be calculated as:

45 × (365/365) × (364/365) × (363/365) × ... × (357/365) ≈ 0.111622.

This result is slightly different from the rough estimate, showing the importance of precise calculations in probability theory.

Exploring the Probability of Exactly Two People Sharing the Same Birth Month

To find the probability that exactly two people out of six share the same birth month, and no other pairs share a birth month, we can follow a different approach. Let's start by calculating the probability that no one shares the same birth month among six people:

The probability is the product of the following:

First person: 12/12 Second person: 11/12 Third person: 10/12 Fourth person: 9/12 Fifth person: 8/12 Sixth person: 7/12

Thus, the probability that no one shares the same birth month is:

(12/12) × (11/12) × (10/12) × (9/12) × (8/12) × (7/12) ≈ 0.00467.

The probability that at least two people share the same birth month is the complement of the above probability:

1 - 0.00467 ≈ 0.99533.

Now, we need to calculate the probability that exactly two people share the same birth month and no others do. This includes two scenarios:

Scenario 1: No Third Person Shares the Birth Month

If we have a pair and no third person shares the birth month, we calculate the probability as follows:

The probability that the third person does not share the same birth month with the pair is 11/12, the fourth person 11/12, the fifth 11/12, and the sixth 11/12. Therefore, the probability is:

(11/12) × (11/12) × (11/12) × (11/12) ≈ 0.3756.

This probability must be subtracted from the overall probability of at least two people sharing the same birth month to find the probability of exactly two people sharing the birth month:

0.99533 - 0.3756 ≈ 0.61977.

Scenario 2: Another Pair Does Not Share the Same Birth Month

If after the first pair no other pair shares a different birth month, we calculate the probability as follows:

The probability that the third person does not share the same birth month with the first pair is 11/12, the fourth 10/11, the fifth 9/11, and the sixth 8/11. Therefore, the probability is:

(11/12) × (10/11) × (9/11) × (8/11) ≈ 0.5409.

This probability must also be subtracted to get the final probability:

0.61977 - 0.5409 ≈ 0.07887.

Therefore, the final probability that among six people, exactly two people will share the same birth month and no others will share a birth month is approximately 0.07887, or 7.887%.

Conclusion

In conclusion, we explored different methods to calculate the probability of finding at least two people born in the same month in a group. From combinatorial methods to approximate calculations, we found that the probability is significantly higher than one might initially think. For a group of six people, the exact probability of exactly two people sharing the same birth month is approximately 7.887%, while the general probability of at least two people sharing the same birth month is about 99.533%.

Understanding these probabilities can be crucial in various fields, including data analysis, statistics, and even in planning events or distributing resources. Whether you are a statistician, a data scientist, or simply curious about the laws of probability, this knowledge can be both enlightening and practical.