Probability with permutations and combinations. Video transcript 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.

## What happened on my birthday?

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. 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?

## Number one song on my birthday?

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. 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.

### Feeling the weight of others’ expectations

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. So if we figure out the probability that everyone has the same birthday we could subtract it from So let's see.

We could we just rewrite this. 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. I also notice the 13th of every month is low.

I always thought your birthday was just whatever day it was but it seems there is way more control than I thought. I had a classmate whose sister was born on September 11, More details here:. Some women offered dates for planned c-section may be less likely to choose the 13th of the month, or Friday the 13th. As for the increase in births starts in June July time.

I wonder if there is a corresponding increase in prescriptions of antibiotics in the Autumn and start of Winter? It would be interesting to see the graphic split for planned c-sections, and another for all other births vaginal delivery, emergency c-section, any induction.

Something else that might be interesting to know — is there a time of year where it is more likely for IVF embryos to be transferred back and similar other fertility treatment to be carried out? Being under consultant care it is more likely babies will be born by c-section. I was born on April 1, but it was in , so if they could calculate the birthdays from , that would be great. Yes, this is surprising, because with my friends group we hate november because 12 of them of 28 born on those dates.

Five on I have thought for some time that low birth rates for major holidays are due to lack of elective c sections. I think doctors and hospital staff would be difficult to arrange for Christmas day, Christmas eve, New year day, independence day. That seems obvious! Would be nice to see the same data set charted to moon phase. And to see if similar trends hold true in other countries. Maybe it is an American quirk to abstain from birth during festive occasions. I should be finished soon if you want to check it out:. Love this! Great work here and what interesting insights we can get from looking at the data this way.

Any plans to expand to include earlier decades? I wonder how different this map is for firstborn vs others. The ability to control delivery — for the benefit of the physician as well as the parents — has to account for some of the weirder anomalies. Obviously babies are being pushed off holidays and being done early induction or later C-sections regardless of the conception date no one plans that well! The dearth of births on all of the 13ths of the months is bizarre.

Is triskaidekaphobia actually that strong? Low birth rates on the days around the fourth of july and Christmas and New Years days. This probably reflects low rates of caesarean sections or the preference of obstetricians NOT to operate on those days they have the day off. This also points to the fact that many caesareans are NOT a medical necessity at the time they are performed. I was supposed to be born on Thanksgiving, so that would have been weird. You have my permission. Hello, Kevin I totally agree with you. People should celebrate their birthdays. Nowadays everyone is so depressed!

We should all try to find something beautiful in our lives. We often depend on other people relatives, friends , so this one day in year should be a special treat!

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