Shared Birthdays Find the probability that of 25 randomly selected people, at least 2 share the same birthday.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the probability that at least two people out of 25 randomly selected individuals share the same birthday. This is a classic application of the 'Birthday Problem' in probability theory.

Step 2: Use the complement rule. Instead of directly calculating the probability of at least two people sharing a birthday, calculate the probability that no two people share a birthday (the complement event). Then subtract this value from 1 to find the desired probability.

Step 3: Assume there are 365 days in a year (ignoring leap years). For the first person, there are 365 possible birthdays. For the second person, to avoid sharing a birthday with the first, there are 364 possible birthdays. For the third person, there are 363 possible birthdays, and so on.

Step 4: Calculate the probability that no two people share a birthday. Multiply the probabilities for each person: $ P(\text{no shared birthdays}) = \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \ldots \times \frac{341}{365} $. This product represents the probability that all 25 people have unique birthdays.

Step 5: Subtract the result from 1 to find the probability of at least two people sharing a birthday: $ P(\text{at least two share a birthday}) = 1 - P(\text{no shared birthdays}) $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In the context of the birthday problem, it quantifies the chance that at least two individuals in a group share the same birthday, which can be counterintuitive due to the large number of possible combinations.

Complementary Events

Complementary events are pairs of outcomes where one event occurs if and only if the other does not. In the birthday problem, instead of directly calculating the probability of at least two people sharing a birthday, it is often easier to calculate the probability that no one shares a birthday and then subtract this from 1.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and arrangements of objects. In the birthday problem, it helps determine the number of ways to assign birthdays to individuals, which is crucial for calculating the probabilities involved in determining shared birthdays among a group.

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