Birthday Problem What is the probability that at least 2 people in a group of 12 people have the

step 1 So this is a fantastic question. In a group of 12 people, what is the probability that at least

Birthday Problem What is the probability that at least 2...

Key Concepts

Complementary Probability

This concept involves computing the probability of an event by subtracting the probability of its complementary event from one. In the birthday problem, instead of directly calculating the probability that at least two people share a birthday, it is easier to calculate the probability that no one shares a birthday and then subtract this value from one.

Multiplication Principle

This principle is used to count the number of outcomes in a sequence of events by multiplying the number of ways each event can occur. In the context of the birthday problem, it is applied to sequentially determine the probability that each subsequent person has a unique birthday compared to those already considered.

Uniform Distribution and Independence

The problem assumes that birthdays are uniformly distributed across all days of the year and that the birthday of each person is independent of others. This assumption simplifies the calculation by treating every day as equally likely and each person's birthday assignment as an independent event.

Birthday Paradox

This refers to the counterintuitive result that in a relatively small group, the probability of at least two people sharing the same birthday is surprisingly high. It demonstrates how human intuition about probability can be misleading, and it highlights the importance of combinatorial methods in probability analysis.

Transcript

00:01 So this is a fantastic question.

00:03 In a group of 12 people, what is the probability that at least two people have the same birthday? well, here's the philosophy of this.

00:11 There are so many different combinations that could occur of two people having the same birthday, like two people could, three people could, four people could.

00:19 And you're talking about two people.

00:21 Maybe it's person one in person 12, person two in person six.

00:24 There's so many combinations that trying to calculate this directly is, i don't want to say impossible.

00:30 But you'd have to have some serious time, software, and mathematical ability to really organize all this together.

00:36 We're going to do a much easier way.

00:39 We're going to work with the complement, and we're going to take the number one 100%, and we're going to subtract from that the probability that nobody has the same birthday.

00:51 So if we do one minus the probability of no common birthdays, that means we are going to find the probability.

01:01 That somebody has a common birthday and it could be two people, three people, four people, et cetera.

01:07 So i know you might think i'm crazy like, oh, this is the easy way, huh? find the probability that there's no common birthdays is really simple.

01:14 I'm not saying that, but at least it's certainly possible in a short amount of time, especially given a graphing calculator or some type of calculator that can handle a little bit of exponent and some big numbers.

01:27 So let's start with person one, right? i'm an analyze.

01:31 Up doesn't matter when their birthday is.

01:33 They could have any birthday.

01:35 365 days, they could have any of those 365 possibilities, but instead of making this even more confusing to start, let's just call that thing one.

01:45 Now, we want to find the probability that the next person doesn't have their birthday in common.

01:52 So whatever birthday that first person has, there's 364 days left out of 365.

01:59 So there's not a very good possibility, people have the same birthday.

02:02 Now we want to go to person three.

02:05 Well, there's already two birthdays, person ones and person twos.

02:08 So there's 363 birthdays left out of 365...

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