Textbook Question
In Exercises 19-22, determine whether the events are independent or dependent. Explain your reasoning.
22. Getting high grades and being awarded an academic scholarship
4. Probability
Multiplication Rule: Independent Events
4:34 minutesProblem 4.2.32
Textbook Question
Same Birthdays If 25 people are randomly selected, find the probability that no 2 of them have the same birthday. Ignore leap years.
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding the probability that no two people in a group of 25 have the same birthday. This is a classic 'birthday problem' in probability, and we assume there are 365 days in a year (ignoring leap years).
Step 2: Recognize that this is a problem of permutations. If no two people share the same birthday, the first person can have any of the 365 days as their birthday, the second person can have any of the remaining 364 days, the third person can have any of the remaining 363 days, and so on, until the 25th person.
Step 3: Write the total number of ways to assign birthdays to 25 people without restriction. Since each person can have any of the 365 days, the total number of possible birthday assignments is \( 365^{25} \).
Step 4: Write the number of favorable outcomes where no two people share the same birthday. This is given by the product \( 365 \times 364 \times 363 \times \ldots \times (365 - 24) \), which can also be written as \( \frac{365!}{(365 - 25)!} \).
Step 5: Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes. The probability is \( P = \frac{365 \times 364 \times 363 \times \ldots \times (365 - 24)}{365^{25}} \). Simplify this expression to find the final probability.
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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 this context, it helps quantify the chance that no two people among a group share the same birthday. Understanding basic probability principles, such as the total number of outcomes and favorable outcomes, is essential for solving the problem.
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Combinatorics
Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. In the birthday problem, it is used to calculate the number of ways to assign unique birthdays to each person in a group. This involves understanding permutations and combinations, which are crucial for determining the total possible arrangements of birthdays.
Complementary Events
Complementary events are pairs of outcomes where one event occurs if and only if the other does not. In this scenario, instead of directly calculating the probability that no two people share a birthday, it can be easier to calculate the probability that at least two people do share a birthday and subtract that from 1. This approach simplifies the calculations and provides a clearer path to the solution.
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