WEBVTT - Short Stuff: Birthday Paradox 0:00:04.280 --> 0:00:06.560 Hey, and welcome to the Short Stuff. I'm Josh, and 0:00:06.559 --> 0:00:08.920 there's Chuck. Jerry's hanging out and Dave is here in 0:00:09.000 --> 0:00:12.160 spirit as always and this is short stuff. And Chuck, 0:00:12.200 --> 0:00:15.520 I have a question for you about this one. Why 0:00:15.520 --> 0:00:20.599 would you do this to us? Why it gets this math? Yes, 0:00:20.760 --> 0:00:24.800 it's not just math. It's famously incomprehensible math. I'm going 0:00:24.840 --> 0:00:26.920 to talk about it and explain it, so thank you 0:00:26.960 --> 0:00:29.920 for that. Yeah, I will say that uh Lori L. 0:00:30.000 --> 0:00:32.600 Dove from house stuff works dot com did a pretty 0:00:32.600 --> 0:00:36.080 good job of of explaining it, I think. Um, but 0:00:36.200 --> 0:00:38.400 I picked this because let me tell you a little 0:00:38.400 --> 0:00:45.240 story real quick, since it's short stuff. Flashback to um 0:00:45.280 --> 0:00:50.640 seven and a half years ago. Okay, allow me when 0:00:51.200 --> 0:00:53.880 uh Emily and I were waiting on our daughter to 0:00:53.960 --> 0:00:57.520 be born. We were adopting her and just she was 0:00:57.600 --> 0:01:00.680 late and late and late, and I was like, jeez, 0:01:00.680 --> 0:01:03.000 when is this kid going to come out? And finally 0:01:03.120 --> 0:01:06.560 when she was born, I was like, oh, I'm curious 0:01:06.720 --> 0:01:10.760 what celebrities she shares a birthday with. And you know, 0:01:10.800 --> 0:01:12.039 I had a lot in my mind at the time, 0:01:12.080 --> 0:01:14.280 so I wasn't thinking if it I knew anyone personally 0:01:14.680 --> 0:01:17.760 and I went to Celebrity Birthdays dot com or whatever 0:01:17.880 --> 0:01:24.120 the website is, and I saw your face, and two 0:01:24.160 --> 0:01:27.399 things happen. Three things happen. Uh. The first thing that 0:01:27.440 --> 0:01:31.639 happened was you gotta be kidding me seriously. The second 0:01:31.680 --> 0:01:35.600 thing that happened was, oh, that's actually really great because 0:01:35.640 --> 0:01:39.520 I'll never forget Josh's birthday like my whole life, and 0:01:39.560 --> 0:01:41.760 it's kind of cool that you guys share a birthday. 0:01:42.400 --> 0:01:44.520 And then the third thing was what is Josh doing 0:01:44.560 --> 0:01:47.800 on Celebrity Birthdays dot com? And why am I not 0:01:47.920 --> 0:01:50.600 on it? Well, my friend, I have an update for 0:01:50.640 --> 0:01:52.760 you because you told that story not too long ago 0:01:53.000 --> 0:01:55.960 and it got me into action. So I used whatever 0:01:56.040 --> 0:01:59.400 cloud I might have at Famous Birthdays dot com and 0:01:59.560 --> 0:02:01.880 um dominated you to be on the site as well. 0:02:01.960 --> 0:02:05.600 So hopefully I'm hoping that they will listen and then 0:02:05.680 --> 0:02:08.519 get you up there on before your birthday. So that 0:02:08.560 --> 0:02:10.720 could be even more embarrassing and more egg on my 0:02:10.760 --> 0:02:14.560 face if they go, no, you deserve it. I even said, 0:02:14.600 --> 0:02:16.720 I was like, he's at least as famous as I am. 0:02:16.720 --> 0:02:18.640 If I'm from there, he should be on there too, 0:02:18.720 --> 0:02:20.760 So it just seems right, you know. So you guys 0:02:20.760 --> 0:02:24.280 share birthday, which is very cool and awesome and fun, 0:02:24.480 --> 0:02:26.680 and I just think it's lovely. Now. Even though it's 0:02:26.720 --> 0:02:30.840 initially like what because you don't want to, like I 0:02:30.840 --> 0:02:33.000 don't know something about sharing birthdays, some people can get 0:02:33.000 --> 0:02:35.600 a little selfish, be like I want my birthday to myself. 0:02:36.960 --> 0:02:39.520 But what we're talking about is sharing birthdays. And what 0:02:39.560 --> 0:02:42.800 are the odds of sharing birthdays with someone? You would 0:02:42.800 --> 0:02:46.600 think it would be one in three d and sixty Yeah, 0:02:46.720 --> 0:02:49.840 And actually I think if you, um put two people 0:02:49.880 --> 0:02:53.040 in a room together, that is the odd although I'm 0:02:53.080 --> 0:02:56.040 sure I'm wrong about it right out of the gate. No, 0:02:56.840 --> 0:03:00.000 I am wrong. I was. There's a one in three 0:03:00.080 --> 0:03:04.000 hundred and sixty four chance if you put two people 0:03:04.200 --> 0:03:07.400 in a room together. The thing is, um, if you 0:03:07.440 --> 0:03:11.280 start putting more people in the room together, the chances 0:03:11.320 --> 0:03:14.160 don't increase linearly. It's not if you put three people 0:03:14.160 --> 0:03:16.600 in the room. It's not like there's a three in 0:03:17.080 --> 0:03:23.240 three hundred and sixty four chance. Man, math, It's not 0:03:23.320 --> 0:03:26.280 like it just increases linearly, like one after the other 0:03:26.320 --> 0:03:30.400 after the other. It starts to increase exponentially, and what 0:03:30.520 --> 0:03:32.960 you end up with is what's called the birthday paradox, 0:03:33.000 --> 0:03:35.440 which if you say that to anyone who knows anything 0:03:35.440 --> 0:03:37.440 about math, they will laugh at you and say, it's 0:03:37.440 --> 0:03:39.839 not actually a paradox. It's just that most people don't 0:03:39.880 --> 0:03:43.360 understand it. We really call it the birthday problem. Yeah, 0:03:43.400 --> 0:03:46.080 because here's the thing. And the more you read about this, 0:03:46.120 --> 0:03:48.840 and the more mathematicians you talk to, they all kind 0:03:48.840 --> 0:03:50.920 of very like they kind of pat you on the 0:03:50.920 --> 0:03:53.520 head and laugh a little bit and say, oh, you 0:03:53.680 --> 0:04:00.440 norms are not very good at calculating things exponentially correct 0:04:00.520 --> 0:04:02.720 like we are. We are very good at it because 0:04:02.720 --> 0:04:04.680 we have studied it and trained to do so. But 0:04:05.320 --> 0:04:08.800 you people, just your little p brains just don't think 0:04:08.840 --> 0:04:11.520 that way, and so you do very rudimentary math that 0:04:11.640 --> 0:04:14.240 is completely wrong when it comes to figuring out like 0:04:14.480 --> 0:04:16.800 the odds of sharing or odds of a lot of things, 0:04:16.839 --> 0:04:19.760 but the odds of sharing your birthday, right and there. 0:04:19.880 --> 0:04:21.640 It's true. They don't have to say it, but it 0:04:21.760 --> 0:04:23.839 is true. It is true. I say, we take a 0:04:23.839 --> 0:04:25.479 break and then we come back and explain what the 0:04:25.520 --> 0:04:50.120 heck is going on here? How about that? Let's do it, okay, Chuck, 0:04:50.160 --> 0:04:52.760 So we should set up the birthday problem or birthday 0:04:52.760 --> 0:04:56.040 paradox to you and me, The question is this, how 0:04:56.160 --> 0:05:00.279 large is a group of people, random people where every 0:05:00.400 --> 0:05:03.880 day of the year, excluding leap years, has an equal 0:05:03.960 --> 0:05:08.640 chance of, um being somebody's birthday, and there are no twins, 0:05:09.120 --> 0:05:12.320 it's all individual people. How many people do you have 0:05:12.400 --> 0:05:16.039 to get in the group before two of them will 0:05:16.080 --> 0:05:22.360 share a birthday? That's right? Wow, did you do that 0:05:22.400 --> 0:05:26.000 off of off of your dome? Know that that's the answer. 0:05:27.120 --> 0:05:29.840 The larger the group you have, the greater the odds 0:05:29.880 --> 0:05:35.880 are obviously, um. So it Yeah, it's an exponential math problem, 0:05:36.000 --> 0:05:39.720 and our brains don't generally think that way. So what 0:05:39.760 --> 0:05:41.600 you have to do is you have to look at 0:05:41.600 --> 0:05:43.560 the number of people in a room. Let's say you 0:05:43.560 --> 0:05:47.320 got your twenty three people, and if you're comparing just 0:05:47.480 --> 0:05:52.599 yourself to the under other twenty two people in the room, 0:05:52.720 --> 0:05:55.800 then you're just gonna end up with those twenty two comparisons. 0:05:56.200 --> 0:05:59.160 But when you're talking about exponential math, you can't just 0:05:59.200 --> 0:06:01.520 look at the one person in that room. You have 0:06:01.600 --> 0:06:05.880 to compare that probability for all the people in the room. 0:06:05.880 --> 0:06:09.039 So the first person would say, all right, I have 0:06:09.120 --> 0:06:12.040 those twenty two comparisons. Then the next person would step 0:06:12.120 --> 0:06:14.880 up and do the comparison, but there would be one 0:06:14.960 --> 0:06:18.080 less because they've already been compared to the one first person. 0:06:18.920 --> 0:06:20.640 And so on and so on until you get to 0:06:20.640 --> 0:06:26.040 the last person. Yeah. Our syllopsism uh misguides us in 0:06:26.040 --> 0:06:28.039 this case because we fail to think about all the 0:06:28.040 --> 0:06:31.920 other people who connect with other people. Right. So I've 0:06:31.920 --> 0:06:34.799 seen a couple of ways to do this. One way 0:06:35.040 --> 0:06:38.719 is to say, um that if you have twenty three 0:06:38.800 --> 0:06:43.479 people in a in a room, Um, you have twenty 0:06:43.520 --> 0:06:50.640 three people times twenty two um possible pairings. Divide that 0:06:50.760 --> 0:06:54.200 number by two, and what you end up with is 0:06:54.200 --> 0:06:58.640 two and fifty three. Okay, that's a really simple easy 0:06:58.720 --> 0:07:01.520 way that Ted Ed taught me to do it. But 0:07:01.600 --> 0:07:03.440 you have to get to the number. Let me put 0:07:03.480 --> 0:07:05.360 it in a different way, Chuck. For that formula, Let's 0:07:05.360 --> 0:07:09.640 say you have five people. Five people have UM twenty 0:07:09.720 --> 0:07:14.880 possible pairings, right, Okay, because if you if you connect 0:07:14.960 --> 0:07:18.400 each person one time, you're gonna come up with twenty 0:07:18.480 --> 0:07:22.400 possible pairings. But half of those are redundant. Right, So 0:07:22.600 --> 0:07:25.640 connecting A to B in person B two A is 0:07:25.680 --> 0:07:29.920 the same thing. That's why you divide that number by two, right, 0:07:30.200 --> 0:07:33.480 so you got five times four equals twenty divided by two, 0:07:33.640 --> 0:07:38.200 which means you have ten genuinely possible pairings in total. 0:07:39.120 --> 0:07:40.880 Another way to do it, to get to the number 0:07:40.960 --> 0:07:44.640 is you take that one the first comparison, three hundred 0:07:44.640 --> 0:07:46.960 and sixty four to three hundred and sixty five divided 0:07:46.960 --> 0:07:49.280 by three sixty five, and then for the next person, 0:07:49.400 --> 0:07:52.320 three hundred sixty three divided by three sixty five, and 0:07:52.400 --> 0:07:54.800 the next person three sixty two, and so on and 0:07:54.840 --> 0:07:57.320 so forth. And if you do that for twenty three 0:07:57.360 --> 0:08:01.840 different people and you take each to the products of 0:08:01.880 --> 0:08:06.680 those equations, all those little little tiny percentages, and multiply them, 0:08:07.000 --> 0:08:09.840 what you come up with is forty nine point eight 0:08:09.920 --> 0:08:13.000 three percent, which means that what you've just done is 0:08:13.040 --> 0:08:15.960 show that there is a forty nine point three percent 0:08:16.000 --> 0:08:18.240 that they're not going to have a birthday. And then 0:08:18.240 --> 0:08:20.800 you just figure out the inverse of that, and you 0:08:20.840 --> 0:08:23.480 come up with a fifty point seventeen percent chance with 0:08:23.520 --> 0:08:25.960 twenty three people that um two of them are going 0:08:26.000 --> 0:08:28.920 to share the same birthday. And again it's because you're 0:08:28.960 --> 0:08:31.960 not coming up with twenty three comparisons, there's two hundred 0:08:31.960 --> 0:08:35.640 and fifty three comparisons, and of just three hundred and 0:08:35.679 --> 0:08:40.080 sixty five days in a year. That's right, I guess. 0:08:40.120 --> 0:08:43.360 The last part of because there's sort of the third 0:08:43.400 --> 0:08:44.800 way to do it, which I kind of started but 0:08:44.840 --> 0:08:47.520 didn't even really finish, is you know, you make those 0:08:47.520 --> 0:08:50.920 twenty two comparisons that first person does, and then the 0:08:50.920 --> 0:08:54.640 next person makes twenty one comparisons, next person makes nineteen 0:08:55.120 --> 0:08:58.439 again because they've already made those other comparisons, and all 0:08:58.480 --> 0:09:00.679 you do is add those numbers all up, you know, 0:09:01.600 --> 0:09:04.360 so on on so on, and adding those together will 0:09:04.400 --> 0:09:07.559 eventually lead to those two hundred and fifty three comparisons 0:09:08.400 --> 0:09:13.200 or combinations of comparisons. Rather, so there's something that escapes me. 0:09:13.280 --> 0:09:16.200 We just generally explained it well. Although I'm sure there's 0:09:16.240 --> 0:09:19.920 some people out there cringing, laughing, crying, who who know 0:09:20.000 --> 0:09:21.960 about this kind of stuff. They're like, this is just 0:09:22.040 --> 0:09:25.400 the saddest thing I've ever heard. We generally explained it. 0:09:26.840 --> 0:09:30.679 I still don't understand how two hundred and fifty three 0:09:31.760 --> 0:09:36.400 comparisons for a possible pool of three hundred and sixty 0:09:36.440 --> 0:09:42.120 five dates leads to a fifty percent chance for three people. 0:09:44.440 --> 0:09:47.640 It doesn't make sense. I'm just airing a grievance really 0:09:47.640 --> 0:09:50.600 more than anything, I don't understand it at all. Um. 0:09:50.640 --> 0:09:53.640 The upshot of it, though, is that, um, when you 0:09:53.679 --> 0:09:59.080 get to seventy people, the pairings have grown so exponentially that, um, 0:09:59.120 --> 0:10:02.840 there's a ninet greater than a ninety nine percent chance 0:10:02.920 --> 0:10:05.559 that there will be a pair of people that share 0:10:05.600 --> 0:10:09.400 a birthday. Again, though we're talking about more than two thousand, 0:10:10.679 --> 0:10:15.680 um comparisons for three hundred and sixty five days. Why 0:10:15.760 --> 0:10:18.640 is that not like five percent chance that there's going 0:10:18.679 --> 0:10:23.080 to be two people that have the same birthday? Yeah, 0:10:23.760 --> 0:10:28.000 I don't know. Uh. With another kind of cool thing 0:10:28.040 --> 0:10:30.400 that was um that Laurie from the House Stuff Works 0:10:30.440 --> 0:10:33.400 article included, which is just another kind of fun example 0:10:33.440 --> 0:10:38.480 of how exponential growth works is and this is I think, um, 0:10:38.520 --> 0:10:41.320 I think she might have interviewed a mathematician. Yeah, his 0:10:41.400 --> 0:10:43.640 name is Frost. Oh yeah, he was laughing at you 0:10:43.679 --> 0:10:45.920 and me the whole time and he doesn't even know us. Yeah, 0:10:45.960 --> 0:10:47.599 he's the one that was like, yeah, you guys just 0:10:47.640 --> 0:10:50.760 aren't very good at this. Uh. Is if you're like, 0:10:50.800 --> 0:10:52.760 if you think of it in terms of money, and 0:10:52.960 --> 0:10:55.560 the example that he used is, um, if you're going 0:10:55.600 --> 0:10:58.160 to be offered a one penny on the first day, 0:10:58.520 --> 0:11:01.760 then two pennies on the second, three pennies on the third, 0:11:02.400 --> 0:11:05.200 and then so on so on for thirty days. It 0:11:05.280 --> 0:11:07.320 might not seem like much money, but at the end 0:11:07.360 --> 0:11:12.000 of the thirtieth day, that is ten point seven million dollars. Right. 0:11:12.200 --> 0:11:14.560 Millionaires who are good at math love to do that 0:11:14.600 --> 0:11:16.760 to people because they turned down this good deal, and 0:11:16.760 --> 0:11:18.480 then they explained to them how it was a great 0:11:18.520 --> 0:11:21.000 deal and they're so dumb. That's how the robber barons 0:11:21.040 --> 0:11:25.120 hoodwinked the generation of people. That's right. Uh, can we 0:11:25.200 --> 0:11:28.880 please end this torment? Yeah, I'm done, Okay, Choice stuff 0:11:28.920 --> 0:11:34.280 is out. Stuff you Should Know is a production of 0:11:34.280 --> 0:11:37.600 I Heart Radio. For more podcasts my heart Radio, visit 0:11:37.640 --> 0:11:40.439 the i heart Radio app, Apple Podcasts, or wherever you 0:11:40.520 --> 0:11:41.800 listen to your favorite shows.