Same Birthdays If 25 people are randomly selected, find the pr...

Question

Same Birthdays If 25 people are randomly selected, find the probability that no 2 of them have the same birthday. Ignore leap years.

Accepted Answer

Welcome back, everyone. If 5 students are randomly assigned a locker each, what is the probability that no 2 students receive the same locker considering the school has 20 lockers, each with a unique code from 1. 20 0.8125 B 0.6750 C 0.5814 and D 0.3921. For this problem we're looking at a probability of an event A. We want to ensure that no two students receive the same locker. We have to recall that. Probability is the ratio of two numbers, A and B, where A is the number of favorable outcomes. And B is the number of total outcomes. So let's begin with B. What is the number of ways that we can assign each student a locker? Well, essentially we have 20 lockers. And We have 5 students. So basically what we can do is visualize each student as a box and understand that each student can receive 1 out of 20 lockers, right? So the first one can receive 20 lockers, the second one, the 3rd 1. The 4th 1 and the 5th 1. So in this context, we're allowing repetition to identify the total number of outcomes, and that's going to have a form of N raise to the power of K. Now N is the number of lockers and K is the number of students. So we get 20 the power of 5. Essentially, we can derive this using the multiplication rule, 20 multiplied by 20, and so on, which gives us 20 races to the power of 5 possible ways to assign 20 lockers to 5 students. Now for the value of a we have to be careful because we want to ensure that no two students receive the same locker which corresponds to permutations. Each student is labeled, right? It's an individual student, so both order and. No repetition or relevance. This is why we're using the number of permutations. So we're going to calculate the number of permutations with um. And K values that are equal to 20 or N and the number of The number corresponding to K is going to be 5, right, which can also be written as 20-factorial divided by 20 minus. 5 pictorial. Which is simply 20 factorial divided by 15 factorial, and that's 20 multiplied by 1918, 17, and 16. So essentially, this can also be derived, right? The first student can get one out of the 20 available lockers, which leaves the second student with the remaining 19. The 3rd student can choose out of the remaining 18 and so on. This means that the probability of an event A is going to be the ratio between A and B. So, the number of permutations with N of 20 and K of 5 divided by 20 to the power of 5, which is going to be equal to. 20 multiplied by 19, multiplied by 17 multiplied by 17 multiplied by 16 divided by 20 to the power of 5, and this is going to be equal to 0.5814, which corresponds to the answer to C. Thank you for watching.

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