Birthday Problem What is the probability that at least 2 people in a group of 35 people have the

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

Key Concepts

Birthday Paradox

The birthday paradox refers to the counterintuitive probability problem which shows that in a relatively small group, the likelihood of at least two people sharing the same birthday is high. This paradox arises due to the rapidly increasing number of possible pairs as the group size increases.

Complementary Counting

Complementary counting is a problem-solving strategy in probability where instead of calculating the probability of the event directly (in this case, at least one matching birthday), the probability of the complementary event (all birthdays being unique) is computed first and then subtracted from one.

Independence Assumption

The independence assumption means that each individual's birthday is considered an independent event, uniformly distributed over 365 days. This assumption simplifies the calculations by allowing the use of multiplication for sequential probabilities without considering dependency between events.

Combinatorics and Permutations

Combinatorics, and specifically the use of permutations, is integral to solving the birthday problem. It involves calculating the number of unique ways to assign different birthdays to people, typically by sequentially reducing the number of available days for each subsequent person, and is essential for determining the probability of no shared birthdays.

Transcript

00:01 In a group of 35 people, what is the probability that at least two people have the same birthday? this is a question where we are going to use the complement.

00:10 We're going to do one minus the probability that nobody has the same birthday.

00:17 One minus the probability of no common birthdays.

00:26 Because if nobody has a birthday in common and somebody has a birthday in common, that's got to be 100%.

00:33 So it means there's 100 % chance either there's no common birthdays or some common birthdays.

00:38 So if we take away no common birthdays, that means the probability left is somebody has to have a common birthday.

00:43 And that could be at least two, three, four, et cetera.

00:46 It covers all those cases.

00:48 So here's how we're going to do this probability.

00:51 We need to know the number of different possible non -birthdays.

00:58 So if we take the first person, they have 365 possible birthdays they could have, there's nobody to have one in common with.

01:05 The next person has 364 possibilities because we don't want them to match up with the other.

01:11 So there's one birthday they can't have.

01:13 For the next person, there's two birthdays they don't want to match up with.

01:16 So we take that down by two.

01:18 But notice for the first, the second, the third person.

01:21 Now, for the fourth person, there's three birthdays that they can't have.

01:26 So that brings them down to 362 possibilities.

01:29 And we've got to do this all all the way out to 35 people.

01:34 So it's really important we notice the pattern.

01:37 For the second person, it's 365 minus 1.

01:41 For the third person, it's 365 minus 2.

01:45 For the fourth person, it's 365 minus 3.

01:49 And for the 35th person, it's going to be 365 minus 34.

01:56 So if we do 365, right, at 35th person, or 35th person can't share our birth.

02:02 With any of those first 34 people.

02:05 So if we do subtract that, that's going to give us a value actually of 331.

02:11 And this is going to be our gigantic numerator.

02:16 Well, how many total possibilities are there? well, there's 365 possibilities for the first person...

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