Birthday Probability Problem The probability that at least 2 people in a room of 30 share the same birthday.
Birthday Probability Problem
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- One of you all sent a fairly interesting problem, so I
- thought I would work it out.
- The problem is I have a group of 30 people, so
- 30 people in a room.
- They're randomly selected 30 people.
- And the question is what is the probability that at least 2
- people have the same birthday?
- This is kind of a fun question because that's the size
- of a lot of classrooms.
- What's the probability that at least someone in the classroom
- shares a birthday with someone else in the classroom?
- That's a good way to phrase as well.
- This is the same thing as saying, what is the probability
- that someone shares with at least someone else.
- They could share it with 2 other people or 4 other
- people in the birthday.
- And at first this problem seems really hard because there's
- a lot of circumstances that makes this true.
- I could have exactly 2 people have the same birthday.
- I could have exactly 3 people have the same birthday.
- I could have exactly 29 people have the same birthday and all
- of these make this true, so do I add the probability of each
- of those circumstances?
- And then add them up and then that becomes really hard.
- And then I would have to say, OK, whose birthdays
- and I comparing?
- And I would have to do combinations.
- It becomes a really difficult problem unless you make kind
- of one very simplifying take on the problem.
- This is the opposite of-- well let me draw the
- probability space.
- Let's say that this is all of the outcomes.
- Let me draw it with a thicker line.
- So let's say that's all of the outcomes of my
- probability space.
- So that's 100% of the outcomes.
- We want to know-- let me draw it in a color that won't
- be offensive to you.
- That doesn't look that great, but anyway.
- Let's say that this is the probability, this area right
- here-- and I don't know how big it really is,
- we'll figure it out.
- Let's say that this is the probability that someone
- shares a birthday with at least someone else.
- What's this area over here?
- What's this green area?
- Well, that means if these are all the cases where someone
- shares a birthday with someone else, these are all the area
- where no one shares a birthday with anyone.
- Or you could say, all 30 people have different birthdays.
- This is what we're trying to figure out.
- I'll just call it the probability that
- someone shares.
- I'll call it the probability of sharing, probability of s.
- If this whole area is area 1 or area 100%, this green area
- right here, this is going to be 1 minus p of s.
- This is going to be 1 minus p of s.
- Or if we said that this is the probability-- or another way we
- could say it, actually this is the best way to think about it.
- If this is different, so this is the probability
- of different birthdays.
- This is the probability that all 30 people have
- 30 different birthdays.
- No one shares with anyone.
- The probability that someone shares with someone else plus
- the probability that no one shares with anyone-- they all
- have distinct birthdays-- that's got to be equal to 1.
- Because we're either going to be in this situation or we're
- going to be in that situation.
- Or you can say they're equal to 100%.
- Either way, 100% and 1 are the same number.
- It's equal to 100%.
- So if we figure out the probability that everyone has
- the same birthday we could subtract it from 100.
- So let's see.
- We could we just rewrite this.
- The probability that someone shares a birthday with someone
- else, that's equal to 100% minus the probability that
- everyone has distinct, separate birthdays.
- And the reason why I'm doing that is because as I started
- off in the video, this is kind of hard to figure out.
- You know, I can figure out the probability that 2 people have
- the same birthday, 5 people, and it becomes very confusing.
- But here, if I wanted to just figure out the probability that
- everyone has a distinct birthday, it's actually a much
- easier probability to solve for.
- So what's the probability that everyone has a
- distinct birthday?
- So let's think about it.
- Person one.
- Just for simplicity, let's imagine the case that we only
- have 2 people in the room.
- What's the probably that they have different birthdays?
- Let's see, person one, their birthday could be 365 days
- out of 365 days of the year.
- You know, whenever their birthday is.
- And then person two, if we wanted to ensure that they
- don't have the same birthday, how many days could
- person two be born on?
- Well, it could be born on any day that person
- one was not born on.
- So there are 364 possibilities out 365.
- So if you had 2 people, the probability that no one
- is born on the same birthday-- this is just 1.
- It's just going to be equal to 364/365.
- Now what happens if we had 3 people?
- So first of all the first person could
- be born on any day.
- Then the second person could be born on 364 possible
- days out of 365.
- And then the third person, what's the probability that
- the third person isn't born on either of these
- people birthdays?
- So 2 days are taken up, so the probability is 363/365.
- You multiply them out.
- You get 365 times 36-- actually I should rewrite this one.
- Instead of saying this is 1, let me write this as-- the
- numerator is 365 times 364 over 365 squared.
- Because I want you to see the pattern.
- Here the probability is 365 times 364 times 363 over
- 365 to the third power.
- And so, in general, if you just kept doing this to 30, if I
- just kept this process for 30 people-- the probability that
- no one shares the same birthday would be equal to 365 times 364
- times 363-- I'll have 30 terms up here.
- All the way down to what?
- All the way down to 336.
- That'll actually be 30 terms divided by 365
- to the 30th power.
- And you can just type this into your calculator right now.
- It'll take you a little time to type in 30 numbers, and you'll
- get the probability that no one shares the same birthday
- with anyone else.
- But before we do that let me just show you something
- that might make it a little bit easier.
- Is there any way that I can mathematically express
- this with factorials?
- Or that I could mathematically express this with factorials?
- Let's think about it.
- 365 factorial is what?
- 365 factorial is equal to 365 times 364 times 363 times--
- all the way down to 1.
- You just keep multiplying.
- It's a huge number.
- Now, if I just want the 365 times the 364 in this case,
- I have to get rid of all of these numbers back here.
- One thing I could do is I could divide this thing
- by all of these numbers.
- So 363 times 362-- all the way down to 1.
- So that's the same thing as dividing by 363 factorial.
- 365 factorial divided by 363 factorial is essentially this
- because all of these terms cancel out.
- So this is equal to 365 factorial over 363 factorial
- over 365 squared.
- And of course, for this case, it's almost silly to worry
- about the factorials, but it becomes useful once we have
- something larger than two terms up here.
- So by the same logic, this right here is going to be equal
- to 365 factorial over 362 factorial over 365 squared.
- And actually, just another interesting point.
- How did we get this 365?
- Sorry, how did we get this 363 factorial?
- Well, 365 minus 2 is 363, right?
- And that makes sense because we only wanted two terms up here.
- We only wanted two terms right here.
- So we wanted to divide by a factorial that's two less.
- And so we'd only get the highest two terms left.
- This is also equal to-- you could write this as 365
- factorial divided by 365 minus 2 factorial 365 minus 2 is 363
- factorial and then you just end up with those two terms
- and that's that there.
- And then likewise, this right here, this numerator you could
- rewrite as 365 factorial divided by 365 minus 3-- and
- we had 3 people-- factorial.
- And that should hopefully make sense, right?
- This is the same thing as 365 factorial-- well 365 divided
- by 3 is 362 factorial.
- And so that's equal to 365 times 364 times
- 363 all the way down.
- Divided by 362 times all the way down.
- And that'll cancel out with everything else and you'd
- be just left with that.
- And that's that right there.
- So by that same logic, this top part here can be written as
- 365 factorial over what?
- 365 minus 30 factorial.
- And I did all of that just so I could show you kind of the
- pattern and because this is frankly easier to type into a
- calculator if you know where the factorial button is.
- So let's figure out what this entire probability is.
- So turning on the calculator, we want-- so let's
- do the numerator.
- 365 factorial divided by-- well, what's 365 minus 30?
- That's 335.
- Divided by 335 factorial and that's the whole numerator.
- And now we want to divide the numerator by 365
- to the 30th power.
- Let the calculator think and we get 0.2936.
- Equals 0.2936.
- Actually 37 if you rounded, which is equal to 29.37%.
- Now, just so you remember what we were doing all along, this
- was the probability that no one shares a birthday with anyone.
- This was the probability of everyone having distinct,
- different birthdays from everyone else.
- And we said, well, the probability that someone shares
- a birthday with someone else, or maybe more than one person,
- is equal to all of the possibilities-- kind of the
- 100%, the probability space, minus the probability that no
- one shares a birthday with anybody.
- So that's equal to 100% minus 29.37%.
- Or another way you could write it as that's 1 minus 0.2937,
- which is equal to-- so if I want to subtract that from 1.
- 1 minus-- that just means the answer.
- That means 1 minus 0.29.
- You get 0.7063.
- So the probability that someone shares a birthday with someone
- else is 0.7063-- it keeps going.
- Which is approximately equal to 70.6%.
- Which is kind of a neat result because if you have 30 people
- in a room you might say, oh wow, what are the odds
- that someone has the same birthday as someone else?
- It's actually pretty high.
- 70% of the time, if you have a group of 30 people, at least 1
- person shares a birthday with at least one other
- person in the room.
- So that's kind of a neat problem.
- And kind of a neat result at the same time.
- Anyway, see you in the next video.
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