Shared Birthdays Find the probability that of 25 randomly sele...

Question

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

Accepted Answer

All right, hello everyone. So this question says, a gym has 50 lockers and each new member is randomly assigned a locker number. What is the probability that in a group of 6 new members, at least 2 will receive the same blocker number? Option A says 0.27, B says 0.35, C says 0.50, and D says 0.06. So for a question like this, we can use the compliment principle, because instead of directly calculating the probability that at least 2 people share the same locker, what we could do instead. I first find the probability of all individuals. Of all individuals getting different locker numbers instead. And so after we find the probability of everybody getting a different locker number, we can then subtract that by 1 to reach our answer. That is the probability that at least 2 people share. So let's do that, right, let's begin with the probability of everyone getting. A different locker number without replacement. All right, so first, you would treat this in order. Starting off with the fact that the first of the 6 new members would have a choice between all 50 of the available lockers. By the time the 2nd member joins, the 1st member would have already selected their locker, which means that the 2nd member has 49 lockers to choose from. And so on and so forth, right, so the 3rd member would have 48 lockers to choose from. The 4th would have 47. The 5th would have 46 lockers to choose from, and the 6th member would have. 45 to choose from. And so for each member, the probability would be. The total number of choices available to them divided by 50, which is the total number of lockers. So the probability of everyone having a different locker number, that is without replacement. Would be 50/50 Multiplied by 49/50. Multiplied by 48/50. Multiplied by 47/50. Multiplied by 46/50. And multiplied by 45/50. And so evaluating this expression gives you a probability of 0.73. So now The final answer That is the probability in which two or more people share a locker would be equal to 1 subtracted by 0.73, which gives you 0.27. And this corresponds to option A, the multiple choice, making that our correct answer. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

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