Question

What is the probability that in a group of 10 people, at least 2 have the same birthday? Assume that nobody was born on February 29 th. Use a calculator to get a good. approximate answer.

Short answer

The probability is approximately 11.7%.

Step-by-step solution

Understanding the Problem

We need to find the probability that at least 2 people in a group of 10 have the same birthday. This is equivalent to finding the probability of NOT all having different birthdays and then subtracting this from 1.

Calculate the Total Possible Birthday Arrangements

Each of the 10 people can be born on any one of 365 days. Therefore, the total number of possible birthday combinations is given by the power of 365, 10 times: \[ 365^{10} \].

Calculate the Favorable Outcome for All Different Birthdays

For the birthdays to all be different, the first person can have any of 365 days, the second person can have 364 choices (as it's different), third 363, and so forth down to the tenth person, who has 356 choices. Thus, the number of ways to have all different birthdays is: \[ 365 \times 364 \times 363 \times \, ... \, \times 356 \].

Calculate the Probability of All Different Birthdays

The probability that all 10 people have different birthdays is the ratio of the number of favorable outcomes (all different) to the total possible outcomes. This can be expressed as: \[ \frac{365 \times 364 \times 363 \times \, ... \, \times 356}{365^{10}} \].

Calculate the Complement Probability

Since we need the probability of at least two people sharing the same birthday, we take the complement of the previous probability: \[ 1 - \frac{365 \times 364 \times 363 \times \, ... \, \times 356}{365^{10}} \].

Solve with a Calculator

Calculate the value of the complement probability using a calculator to get an approximate answer. You should find that this probability is approximately 11.7%.

Key concepts

Probability

Probability is a branch of mathematics that deals with the likelihood or chance of different outcomes. In the context of the Birthday Problem, we are interested in the probability that a specific event occurs: at least two people in a group share the same birthday.
To find this, we first calculate the probability of the complementary event (i.e., that all people have different birthdays) and then subtract this probability from 1.

• If all 10 people have different birthdays, the first person can choose any of the 365 days, the second person one of the remaining 364 days, and so on.
• Multiplying these probabilities together gives the probability that all 10 have different birthdays.
• The complement probability, that at least two share a birthday, is then found by subtracting from 1.

This approach highlights a key principle of probability: often, it's easier to calculate the complement and then find the desired probability.

Combinatorics

Combinatorics is the study of counting, calculating, and arranging possible objects in a set according to specified rules. It's crucial in solving the Birthday Problem as we need to determine the number of possible arrangements of birthdays.
In our problem:

• Each of the 10 people has 365 possible birthdays, leading to \(365^{10}\) potential combinations.
• To ensure all 10 birthdays are different, the number of arrangements reduces as each subsequent person has one less choice: 365 for the first, 364 for the second, down to 356 for the tenth.
• This decreasing sequence results in the product \(365 \times 364 \times 363 \times \ldots \times 356\).

Combinatorial techniques like these are used to enumerate possible outcomes, laying the foundation for calculating probabilities.

Discrete Mathematics

Discrete Mathematics involves distinct and separable values, perfect for analyzing finite structures like those in the Birthday Problem. It encompasses both probability and combinatorics.

• Each person’s birthday is a discrete event, with only a finite number of possibilities (365, excluding leap years).
• To solve the problem, we calculate probabilities through discrete steps: enumerating possible outcomes and applying the principles of combinatorics.
• The method of complements, a common discrete mathematics strategy, was used here to find the probability of at least two shared birthdays.

Understanding discrete structures helps in managing complex problems by breaking them down into countable, manageable parts, essential for practical probability calculations.