23 and Football Birthdays – Numberphile

23 and Football Birthdays – Numberphile


OK, so today we’ve got exclusive
access to Nottingham Forest football ground here. Two-time European Cup winners
who have given us the access so that we can talk about a
very important number in football, and a very important
number in probability, which is the number 23. OK, so 23. So why is 23 an important
number in football? Some of you may know
this already. That’s the number of people
that you will have on the pitch during the game. That’s two teams of 11
and the referee. So there are 23 people
on the pitch. Now, here’s my question today. What’s the probability that two
of those people will share a birthday? The answer may surprise you. So we’re not talking about the
year, we’re just talking about the date itself. Maybe it’s the 14th
of January, maybe it’s the 5th of June. We’re talking about
the date itself. So let’s work this out. I can make this slightly
easier if I ask the opposite question. I’m going to work out the
opposite, which is, what’s the probability that no one on that
pitch shares a birthday. That’s an easier question
to answer. Let’s do this. OK, so your first player. OK, it doesn’t matter what
birthday he has. But when your second player gets
on the pitch, what’s the probability that he doesn’t
share a birthday? Well, he will have, out of the
365 days to choose from, he can have a birthday on 364
of them out of 365. So we’re not including February
the 29th, no leap years here. And we are assuming that all
days are equally likely. OK, so your second player has
to have one of these days. Your third player, when he comes
on to the pitch, will have a choice of 363
days out of 365. And then what next? The fourth player. Out of those remaining days, he
will have 362 out of 365. And you can keep going. Eventually, you’ll get
to the 23rd player. Let’s call him the referee. So your 23rd player, how many
choices does he have? He will eventually get 343
days left out of 365. So this is the number of days
that you’re allowed to have for that referee’s birthday. Because we are looking at no
one sharing a birthday. Now, if you want to find out
the probability that no one shares a birthday, you multiply
all these together. And you’ll get a number. And that number is around
about 0.493. And if you’re not happy with
probabilities like that, that’s 49.3%. Just slightly under half. We were interested in the
opposite question. The opposite question was,
what’s the probability that someone does share a birthday. That’s the opposite thing of
what we’ve worked out. So the probability that someone
does share a birthday will be 50.7%. It’s slightly over a half. You’re more likely for two
players to share a birthday than if they don’t
share a birthday. And that’s quite surprising. And people want to think,
well it must be something like 100 people. You must need 100 people
for that to be true. Or 200 people. If you think of it this way,
think of all the pairs of people you could make out of
23 people on the pitch. All the possible pairs of
people– in fact there are 253 pairs of people you
could make. And, well, think of it that
way– you start to see why it’s quite likely, that
two of those people will share a birthday. So next time you’re at a
football match, think of it– you have a greater than 50%
chance that two of those people share a birthday. OK, so here’s another way
to think about it. Imagine you’re watching
the game. And well, if you need to go to
the toilet, you get up and you have to move past all
the other people. By the time you’ve moved past 23
other spectators, there is a greater than 50% chance that
two of those people you walked past will have shared
a birthday.

About the Author: Garret Beatty

100 Comments

  1. The probability is reduced from around.50729 to around .50686. if leap years are assumed have a 1 in 4 chance of happening in any given year (which isn't really true).

  2. Thanks Brandon A, I know leap years have slightly less than a 1 in 4 chance of occurring in any year (as 3 in every 400 don't happen) but looking at your figures I don't think it will make a significant difference.

  3. I calculated it for what we across The Pond call football ("gridiron," or whatever you prefer). In the NFL, there are 7 officials in addition to the 11 players per team. So with 29 people, it's better than a 2:1 chance, 68.1 % to be, well, specific.

  4. Alternatively, the way I've always understood it, is if you raise the likelihood of two people not sharing a birthday to the power of the number of possible pairs (in this case 253) you will get the 49.95%, or a 50.05% chance that someone shares a birthday.

  5. 23, 23, 23, what does it have to do with CLARA???? We have to wait so long until DW comes back!!!!

  6. Came here after 42, kinda was expecting to see some references to Jim Carrey's film 🙂 Still interesting video.

  7. Hey. This makes me wonder if you guys have done a video on the Monty Hall problem. I haven't looked at all 153 videos yet, so I don't know. But if you haven't, you should! It's really fun.

  8. My math teacher used to always tell us this probability but I never understood it until now. That's pretty amazing. If only my calculus lecturer was as good at teaching as you!

  9. PARADOX: In a group of 23 people there is about 50% chance that there is a shared birthday. So 2 groups of 23 has a theoretical probability of about 100% of a shared birthday. However, in a single group of 46 people, there is only a 94% chance, even with many more possible combinations.

  10. not necessarily true. There are people who grow up speaking multiple languages and they might use those languages interchangeably, possibly while writing things also.

  11. LOGICAL FALLACY DETECTED! By that same reasoning, if you had two groups of 46 people, each with a 94% chance of having a shared birthday, then 2 groups of people would have a theoretical chance of 188% chance of a shared birthday. This is clearly not true and the problem lies with how the math is worked. each additional person you add increases the chance of having a shared birthday but the chances that are being added are getting smaller and smaller so you cannot add two probabilities like that

  12. I just liked this so that:

    Likes = 1,111
    Dislikes = 111

    Let's call this the last "binary popularity" for a while. Next binary popularity would be, what, 10,000 likes if dislikes remains at this? Unlikely… oh, and it was a binary popularity before I liked it 🙂

  13. that isnt right because he said that after you walk past 22 people it is a slightly larger than 50% chance that anybody you walked past shared a birthday with you or another person you walked past.

  14. At first I was totally amazed. But, then, when I thought about it, if you imagine ropes between 23 people, all connected to every other person, if you counted the ropes it would be a large number, around half of 365

  15. Question: Shouldn't the first fraction for odds no-one sharing a birthday be 365 over 365 (100% of not sharing) as there is no-one to share a birthday with?

  16. If you like it better that way, that's fine, since the answer still comes out the same. You could even go a step further and say that before the 365/365, you should start with 1, since no one can share a birthday if there are zero people, either. So, 1 * 365/365 * 364/365 * ….

  17. Very interesting so 50.7% chance that a player or ref on the field will share a birthday. If one person was on the field wouldn't there be a 365/365 chance?

  18. This blows my mind. I never thought that there is such a good chance that I have the same birthday as someone else on the field.

  19. "This blows my mind. I never thought that there is such a good chance that I have the same birthday as someone else on the field."

    As I see it this is why this is so apparently remarkable.

    The actual statement refers to any two people, but I think we're programmed to consider it as "Me + 1 Other Person". The probability of that, of course, is much lower – no idea what though 🙂

    Maybe this video would be better placed in a psychologyphile channel.

  20. im surprised that this is the only video by you guys about the number 23. i mean, theres a blockbuster movie about it. 
    that aside though.. ive always seen the number 343 appear in my everyday life to the point where it got annoying (just like the number 23 in the movie) and ive tried to ignore it.. but you just made me realize the the number of days in a year minus 23 is 343. i guess im not sleeping tonight.

  21. This is mind blowing. I didn't expect that, especially as a soccer player. Recently, we also did probability in my class…. Ahhh the memories. If you were to take the probability of two people in the whole stadium sharing a b-day, how would you work on the problem after you reach 1/365?

  22. This was very interesting, not only as a fan of football, but also as a math student. Probability is a key aspect and important part of our day to day lives. As humans, we continually put ideas, themes, and occurrences into quantitative values. Probability is one of the more favored way to measure aspects of our day to day lives. Realizing the importance and realities of these probabilities is also very important. And as we see in this video, things in our day to day lives are much more probable than we may have otherwise expected. How did the idea of using a full manned pitch as an example come about? What are other ways in which probability can be better applied to our daily lives?
     

  23. You should also take in account the line judges. But, its interesting that you found a way to apply math to the beautiful game. Well done.

  24. This was definitely an interesting application of probability. I do have to wonder how to answer might vary if leap years and the likelihood of birthdays occurring in certain months over others was taken into account. (Which obviously for the purposes of this video was completely unnecessary, but would still be interesting to consider).

  25. How cool! The relation between the game and the math was so interesting! I liked how you showed the players birthdays and showed who shared one. Awesome video!

  26. There seem to be a disproportionate number of players born in January… This looks like something out of Malcolm Gladwell's book Outliers.

  27. This seems counter intuitive but the math never lies! Great video 🙂
    How can I calculate the probability in a group of 23 (or n) people of another person having the same birthday as me? In other words what is the probability of me being one of the two people sharing the same birthday?

  28. I like how you guys related the birthday problem to football to make it more interesting, which works perfectly well if we ignore the various assistant referees (linesmen, 4th & 5th officials, etc). 

  29. I wonder why it happens to be the number 23 that this probability is true. I guess it makes sense when you think that there is an 100% to people share a birthday if there are more than 365 people in a given group. 

  30. I showed this to a statistics class of 21 lower sixth and then checked to see if there was a pair among them. Well, there was a pair of twins. And the first two pupils on the register were a pair. And there were two other pairs. So what is the probability of two pairs, or more pairs? I think we can express it in terms of nCr and factorials. 

  31. This video was shocking to me! I had no idea that the probability of two soccer players sharing a birthday was so high. It at first seems so unlikely. Has there been any data collected on this topic? How often do two people on the pitch share a birthday in real life?

  32. This video was very interesting and left me thinking about that it is more likely for out of 23 people 2 of those people share a birthday. I was confused at the start of the video but by the end numberphile had fully explained why his claim is true

  33. Mmm, seems to me that the formula would be 364/365 x 363/364 x 362/363, etc. Strange, since I have never met someone that I share a birthday with and I am 50 🙂

  34. The number 23 is very important in soccer because that is how many people are on the field at once, including referees. I really enjoyed learning how to find the probability of something, like birthdays. It is quite shocking that is it more likely for two of the 23 players to share a birthday than for all of them to have different birthdays. This video showed me an easy way of how to determine probability. All you do is take the number 364 out of 365 days and multiply it by the second guy which is 363 out of 365 days and so on… Then you add multiply the bottom and the top of the fractions and divide and you will have a decimal. Move the decimal two places to the right to get your answer. Thank you!

  35. My math(s) class (plus a few random kids from the hallway) compared birthdays and all 40 or so of us did not have any birthdays in common! That must be pretty rare! (My teacher bought us pizza for this!)

  36. Its amazing how you can figure out someones birthday just from number 23 and the number of days in a year.

  37. There are 7 000 000 000 people on Earth. There are 365 days in a year. There are 366 days in a leap year. 7 000 000 000 / 366 = 19 125 683 That means that roughly over a 19 million people shares a birthday with you.

  38. I had to watch this video for a school project we had to do and I was truly fascinated by the statistic that it was more likely that 2/23 people were more likely than not to have birthdays on the same day. I really like the way they tied this statistic together with soccer and then later used a specific example of two players that had the same birthday. This is almost as fascinating as the the stat that all three of the Dallas Mavericks point guards (Deron Williams, JJ Barea, and Raymond Felton) were born on the same day. The likeliness of this occurrence is 0.00182%. This means it would take statistically approximately 1270 years for this specific event to happen again.

  39. I am super surprised by the outcome of this video. It made me think hard! I would've never guessed that the probability of two people out of 23 people are more likely to share a birthday (by 50.7%) than it is for them not to share a birthday. It made complete since when the math was being done to show how this was possible. What would be the probability that two people out of a certain number had a birthday in the summer? How could you figure this out? Also, are there any other numbers other than 23 that this works for? Very cool video.

  40. There was a slight mistake in how the problem was defined. He said "the chance that 2 people out of 23 share a birthday" but what he's really calculating is 2 or more sharing a birthday.

  41. I am really suprised by the outcome of this video. It made me question many things and made me think hard. I would've never guessed that the probability of two people out of 23 people are more likely to share a birthday (by 50.7%) than it is for them not to share a birthday. It made sense however when the math was being done to show how this was possible. What would be the probability that two people out of a certain number had a birthday in the summer? How could you figure this out?

  42. This video taught me more about probability and how it can be used in real life situations. Obviously there are many more ways of using probability, but it does put the world in a whole new view. Probability helps give us an idea of the risk of a chance before we take that chance which can be seen as very helpful when making large decisions.

  43. This showed me that math was really important in real life. It taught me that math is a thing that we have to use in real life situations. It showed why we need to learn math and why it is important in life. This video will be helpful to the people that have a question like "why do we learn math if we don't use them in real life?"

  44. I really enjoyed this video, because it shows us that math can be brought in to real life situation, it also showed me how probability can be used to figure a lot more than you actually think. Before watching this video, I would've never thought that there's a 50% chance that two players have the same birthday in every match, but all the rules of probability prove that it is possible. I really enjoyed this video!

  45. To calculate it faster by hand: 364×363×…×343=364!/342! So just type (364!/342!)/365^22 and you get the same result. (your calculator might not be able to handle the factorials so try using wolframalpha)

  46. I am super surprised by the outcome of this video. It made me think very hard! I would've never guessed that the probability of two people out of 23 people are more people are more likely to share a birthday (by 50.7%) than it is for them to not share a birthday. It made complete since when the math was being done to show how this was possible. What wold be the probability that two people out of a certain number had a birthday in the summer? How could you figure this out? Also, are there any other numbers other than 23 that this works for? Very interesting and cool video!

  47. What is the probability that someone watching this video also has the same birthday as the two players! 5th march thats me!

  48. So Basically, in a group of 23 people, there is a 50.7% Chance someone shares a birthday with you and a 49.3% chance that someone does not share a birthday.

  49. Barring any genetic anomalies between contributed egg and sperm, each [human] parent contributes 23 chromosomes to their resulting child when they reproduce

  50. I absolutely love these videos, but the markers were a bad choice. Watching these videos with headphones is just killing me.

  51. Hey so I made a probability-based football video on how likely you are to win the Super 6 jackpot, taking inspiration from these Numberphile videos

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