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Minisode 0.6: Four Problems
428 Plays7 years ago
Jonathan and Gabriel discuss four challenging problems.
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This episode is sponsored by
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Support this podcast: https://anchor.fm/breakingmathpodcast/support
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Transcript
Introduction and Advertisements
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Exploring the Birthday Paradox
00:01:54
Speaker
What we mean by this is that these problems are counterintuitive, but are, by any stretch of logic, correct or incorrect. These problems will include the birthday paradox, a common mathematical fallacy, the shape of Gabriel's trumpet, and the gambler's ruin. We present this episode in the hopes that you enjoy the challenge that they present. We hope you enjoy this minisode.
00:02:25
Speaker
I'm Jonathan. And I'm Gabriel. And what's the first paradox or weird question that we have today? Very first, very first paradox is actually one that I just read about recently. And I suppose it's well known in some circles. It's called the birthday paradox.
00:02:41
Speaker
Yeah, and let me state the exact question. How many people do you have to have in a room in order to have a 50% chance or greater of any two sharing an exact birthday with one another? Exact birthday meaning month and day and not necessarily year.
00:02:58
Speaker
This one absolutely blew my mind. I had my guess. I went about trying to solve it. And then once I saw the solution, I was completely wrong. So I'm hoping to find out if some of our listeners can empathize with me and see if I'm not the only one who gets this wrong.
00:03:16
Speaker
And we're going to show you the math behind it later, but first we want you to use your intuition. Is it 1 out of 365, n out of 365, where n is the amount of people in the room, greater than 100, greater than 50, or greater than 20, or less than 20, I guess. So you have a few options there.
00:03:39
Speaker
That sounded like five options that we're going to give to our listeners, right? And should we just say it one more time? Sure. Is it one out of 365, N out of 365, where N is the amount of people in the room, greater than 100, greater than 50, greater than 20, or less than 20?
00:03:59
Speaker
Hmm, interesting, okay. And of course, you know, in this case, one of the numbers, of course, is within a year there are 365 days. We are not counting leap year here. So now we're going to continue on to the next paradox.
Infinite Series and Mathematical Concepts
00:04:14
Speaker
This one is a simple one. What is the sum of 1 plus negative 1 plus 1 plus negative 1 plus 1 plus negative 1 ad infinitum?
00:04:27
Speaker
Now didn't this, this particular problem that you just mentioned, I think this has an origin with the mathematician Frederick Gauss, is that correct?
00:04:33
Speaker
I believe it does. I would have to look that up to make sure. But if it does, I wouldn't be surprised. Gauss is one of my favorite mathematicians, and this is right up his alley. Interesting. Yeah, that seems like such a simple problem. But again, with the theme of our episode, it may not be what you expect. So again, we're starting from summation from n equals 0 to infinity. Is that correct? Yes. OK.
00:05:00
Speaker
And then it's just one plus negative one, et cetera, et cetera, et cetera. And then what is the final answer? And that is the question.
00:05:09
Speaker
And so we'll give you a while to think on that one. We're not going to go over the answers yet. We'll go ahead and introduce you to the third problem that we're going to discuss on today's Minnesota. The third one is called Gabriel's trumpet. And of course, I'm not referring to myself here, although I was named after this Gabriel. This is the Angel Gabriel. It's called Gabriel's trumpet. And Jonathan, do you want to explain why this problem is called Gabriel's trumpet?
00:05:34
Speaker
Sure. There's a shape called Gabriel's Trumpet. And what it is is you can imagine the graph of one over X. That's to say that at zero, it's infinity. At one, it's one. At two, it's one half. At three, it's one third. So it's one over whatever number it is. And if you're looking at that graph, of course, it looks like a trumpet. It, you know, it starts high and it curves downward and then it asymptotically approaches zero.
00:06:02
Speaker
or at least the profile of the trumpet. And what Gabriel's trumpet really is, is that surface being revolved around the x-axis.
Gambler's Ruin and Game Strategies
00:06:10
Speaker
So let me break this down for some of the people who might have trouble visualizing graphs. You have this shape that goes really high up and then really far down and then almost approaches zero. You take that and imagine a lathe. A lathe is a thing that you make baseball bats on.
00:06:29
Speaker
But imagine you're making a trumpet out of the baseball lathe where the open part is infinitely large and then trails off into this very, very tiny shape. What we're asking about this shape is what's the surface area and what's the volume? This last problem is referred to as the gambler's ruin. And once again, Jonathan, would you like to introduce this one?
00:06:51
Speaker
Sure, I'll introduce it with the simplified version, because the full version is a little bit complex, and I think that it doesn't add anything to the understanding of this. Let's say we're playing a game, and you have a 50% chance of winning every time. If you win, you get double the money, and if you lose, you get none of the money. What's the best way to play this game, and is there any way to play this game with a finite amount of money and not get broke?
00:07:21
Speaker
Interesting okay, so and then basically when we talk about strategies obviously it's it's a simple game of chance and you basically Do a fraction of the money that you have? For each win is that correct?
00:07:35
Speaker
Um, yes. And, but that fraction could be, uh, any, you don't have to invest half of it. You can invest twice as much. You were, we're assuming. So like, if you do one, one dollar on the first one, you could do like $40 on the second one or something like that. We're saying what's the best strategy. And we're going to give you a second to think about that. So if you'd like to work on these problems by yourself, you can pause right now.
00:08:06
Speaker
And we're back. All righty. At this point, we get, we get to go over these, uh, these four problems and you can see if you were correct. So I think the first problem we were talking about is the birthday paradox. Before we give you the answer, let's talk about that problem one more time. Shall we? Yeah. How many people do you have to have in a room in order to have a 50% chance or greater of two people in that room sharing an exact birthday with one another?
00:08:31
Speaker
And I think we actually gave you a choice of four or five options here. The first option I think was, is it a one out of 365 chance? Is that the correct answer? What are the other options? N over 365, where N is the number of people in the room, greater than 100, greater than 50, greater than 20, or less than 20. And the final answer is, drum roll please,
00:09:00
Speaker
Well, before we get that, I'm going to belay your drum roll. I'm going to give you a way of thinking about this problem in case you want to pause one more time and you didn't get it the first time. What's the chance of two people not sharing a birthday? Oh yeah, that'd be a huge, huge chance. So if you just start with just two people in a room, there's a huge chance of them not sharing it.
00:09:23
Speaker
Yeah, in fact, it's 365 over 365 times 364 over 365. And I think that gives you enough of a clue. So if you want to pause, pause now.
00:09:38
Speaker
And for some of you, we are back. For other of you, we are not. And the answer for 365 different days is 23. You only have to have 23 people for there to be a greater than 50% chance of to sharing a birthday.
00:09:54
Speaker
I got that completely wrong the first time I did it. And honestly, I don't think I had a real strategy in mind. I just thought how many people would I have to have for myself in order to have some chance of somebody being born on my birthday. And of course, that was part of my first problem. It's not only about me, it's about any two people at all.
00:10:14
Speaker
Yeah. In fact, the answer is completely different. When you want, if you want to say how many people you would have to have in a room to have the same birthday as you, the expected number would be 365. Yeah. Yeah. But then by the end, it changes when you consider any two people at all. So this is, this is a great lesson. I think, um, I think that, uh, uh, statistics in general are, are very important, you know, for, for all kinds of reasons. And this gives you reason to really carefully consider the information that you're given when somebody is rattling off stats.
00:10:44
Speaker
Now how many hours are there in a year, Gabriel? There are in a year, if we get some quick calculations, I want to say 8,760. And so how many people would it take for two people to share not only the same day of the same month for their birthday, but the same hour of the same day of the same month? Ooh, do you want me to just guess here? I'd be happy to just...
00:11:07
Speaker
has already guessed here. It seems like a very low probability. I know it's 23 was the first answer, but if we make it a little more complicated and they're still out of 24 hours, so we're going to increase it, it's going to be more than 23. I know that. Oh yeah, and we're dealing with a number that's about 30 times as big as 365.
00:11:32
Speaker
Yeah, now I know that you already gave the formula away, so solving it really isn't too tricky. Do you want to say the formula one more time?
00:11:39
Speaker
Yeah, the formula is 365 over 365 times 364 over 365. And you keep doing that until you get 365 minus N over 365. And that total number is less than one half. And then that's the chance of those people not sharing a birthday. So the chance of them sharing a birthday is one minus that. So, um, yeah. And then to add the extra element of sharing the same hour in the day,
00:12:08
Speaker
All you have to do is multiply 365 by 24 and plug in and do the same thing over and over again.
00:12:16
Speaker
Yeah, actually, and the answer to that one is you need 111 people. Can you believe that? So how often are you in a room with 111 other people? You know, like at a small concert or, you know, like maybe a high school play. So there is approximately a 50% chance that you share not only the same birthday, but the same hour as somebody else. In fact, I remember in my high school theater class,
00:12:44
Speaker
There were two people who I shared the same birthday with. That was pretty remarkable. And seeing it from your end, that's even more remarkable from the end of the teacher that's not as remarkable. And that's the interesting thing about statistics is all about perspective. Oh, that's right. That's right. Yeah. Good point. Wow. We should do more probability problems. I like these.
00:13:05
Speaker
And the last one is the exact minute of the exact hour of the exact day of the exact month, 525,600 possibilities. And how many people do you have to have in a room for that Gabriel? We crunch the numbers. To have the exact minute, the exact hour and the exact day, you'd have to have 854 people to have a 50% chance of that happening.
00:13:30
Speaker
That's all. So it does, it does not increase in proportion with the number because that's 60 times as large as the number before. That's like a, that's, that's like what, like a medium to small high school in certain towns, 854 people. Yeah. I mean, yeah, I think I went to a high school with, uh, about 1200 people. Wow. So there was, so there's a greater than 50% chance of two people sharing the exact minute.
00:13:55
Speaker
Wow, that's great. So really, if you just say that you're a superintendent and you visit a whole lot of high schools, there's almost, well, I guess you wouldn't say that there's a guarantee that eventually you will find a person who meets this. In fact, quite the opposite. It's much less likely. Yes. Again, that's a problem of perspective.
00:14:16
Speaker
But the likelihood of finding two people, any two people who share the exact minute, the exact hour, the exact day of any two people, you know, you still only need 854 people. That's quite remarkable. Alright, so should we move on to the second problem?
00:14:36
Speaker
Definitely. So what's one plus negative one plus one plus negative one plus one plus negative one, etc. And the answer I'm going to say is, well, it depends. It does depend because what's one plus negative one? Zero. And then what's one plus negative one? Zero. So then if you keep grouping them like that, do you get zero plus zero plus zero plus zero, which is always what? Zero, of course.
00:14:58
Speaker
However, if you group them the other way, like this, one plus negative one plus one plus negative one plus one plus, et cetera, you get one plus zero plus zero plus zero, which is a final answer of one. So it could either be zero or one. So the solution to this is that these are illogical because they do not converge.
00:15:22
Speaker
Yes, so we threw you guys a curveball and you know, part of me was like, should we apologize to our audience for throwing them a curveball?
00:15:29
Speaker
No, I don't know. I don't think so. No, we shouldn't. And what's interesting about this is that the way that you kind of formalize it is that you have to say there has to be, for every epsilon, meaning like a number epsilon, there has to be a number n such that the nth thing in the sequence and beyond, the difference between any two of those is less than epsilon.
00:15:54
Speaker
Interesting. Now, is there a more modern English way that you could phrase that? Sure. For every little number, tiny number, there has to be a point at which any two subsequent elements of the sequence are less than that tiny number.
00:16:12
Speaker
Now if you were to do one of those, what's the expression if you're reading on Reddit and somebody puts, is it TLDR? Too long, didn't read. The TLDR of this would simply be for some problems, it depends on how you group it.
00:16:29
Speaker
Or you could say that if it depends on how you group it, it's not a valid problem. Indeed, indeed. And then we've got the next problem we introduced you to, which was Gabriel's trumpet. And we already, we talked about the shape of the trumpet, which, you know, if you see a, if you see a trumpet sideways or rather part of a trumpet, the horn part. And then it, what was the, the two, the two bits of information that were
00:16:58
Speaker
Well, the graph is that of one over X, meaning that when you start at the mouth of the trumpet, it's infinitely large. Let's say we're measuring in feet. If you go like seven feet down the trumpet, it's only one seventh of a foot in radius. Okay. Right. And then I think the two things that we're asking for was the surface area and was it the volume? Yep. And the volume is pi. Interesting. Yeah, that's really cool. I like how often pi shows up.
00:17:28
Speaker
Yeah, and you could prove this with some very simple calculus. If the phrase simple calculus seems like an oxymoron to some of you, then just you could take our word for it, or you could find some very, very good explanations online. And then we know that the volume is pi, or pi units, whatever units those might be. Now, it would be volumetric units, right? Yeah, so it would be units cubed.
00:17:54
Speaker
Okay, and it would be pi cubed. It would still just... No, no, it would be pi units cubed. Now, the next question was the surface area. So what is the surface area of this shape? It is infinite. Infinity. So it's quite interesting that you can have something that has a finite volume, but an infinite surface area. That's mathematics for you.
00:18:16
Speaker
And so if you think about it, you could increase the shape of Gabriel's trumpet. Like, let's say you're trying to gold plate it. Are you getting increase the shape of Gabriel's trumpet and get a gold, get a gold plating. But as the farther you go down, the thinner the plating would be. So to plate it evenly, you would have to have an infinite amount of gold leaf.
00:18:38
Speaker
And then we've got one last problem that we introduced you to, and this was the gambler's ruin problem. And this problem was asking you to devise a strategy in a game of chance in order to have the best outcome for you. What were some of the other details of this particular problem?
00:19:00
Speaker
You have a 50% chance of winning, and if you win, they give you twice the amount of money. If you lose, they take all your money. So it's basically a double or nothing with a 50% chance. Okay. Very good. Very good. All righty. And we said, is there a way to have a surefire way of winning this or any gambling challenge?
00:19:22
Speaker
Yes. If you have an infinite amount of money and the casino lets you bet an infinite amount of money at any given or arbitrarily large amount of money at any time, all you have to do. So if you start off with $1, let's say, let's say you lose, right? Okay. So I lose. All right. So you're down $1. So bet $2 and then if you win, you're up $1 then you lose. And if you continue this, the most you can expect to win is what you started with $1.
00:19:51
Speaker
Okay, and that's only if I can bet an arbitrarily large amount and an infinite amount of times. Yes, and what I mean by expect to win is I mean the expected value of your winnings, which is a slight difference. I would look up expected value. It's a very complicated concept in which we don't have time to go into. But one thing that really threw me for a loop when I first heard this problem
00:20:16
Speaker
is I would think, okay, let's say I'm up by four bucks. Why can't that be my starting point? And why can't I expect to win four bucks and then eight bucks and then 16 bucks and so on. And the reason why is because of the law of large numbers. You're always going to get back to where you started. Interesting. So yeah, and actually I'm not even familiar with the law of large numbers myself. So there's something that I can certainly research as well.
00:20:40
Speaker
The simple version of large numbers is that if you flip a coin in infinite amount of times, you're going to get about half and half. Oh, got it. Got it. Okay. And the complicated version is that any, um, uh, is that the sum of any, uh, set of random variables is going to approach a Gaussian, but we're not going to go into that and it that, right. And then the other question I have on this particular problem is, is there a surefire way to, to lose all your money?
00:21:06
Speaker
Yes, have a finite amount to start with because you can be up and down, but there's going to be a point where you're trying to win and you're going to not have enough money to bet double. That's to say you might be down 65,535 bucks and you're trying to bet 65,536 when your original dollar back, but you don't have that. So that's how you can lose and you will eventually always lose.
00:21:36
Speaker
Well, that has been our four challenge problems for this week's minisode. We hope you enjoyed it. And until next time, ruin some gamblers.
00:21:50
Speaker
Don't do that.
Teaser for Upcoming Episode
00:21:52
Speaker
And the minisode's over, but we want to talk to you about our upcoming show. We are absolutely ecstatic about a very, very long podcast episode that we just recorded that we are in the process of editing. We thought we'd give you a nice little teaser about this one. Jonathan, are you, are you excited about this one? I am. We talk about artificial and natural states of consciousness and how they can be altered.
00:22:14
Speaker
Yes, altering states of consciousness. We have a very special guest on the episode. The guest has been diagnosed with a condition schizophrenia, and he really tells the listener a lot about what it's like to have this condition, what he experiences, and also some really insightful things for those who don't have it that we could possibly benefit from knowing.
00:22:40
Speaker
And we also have a guest who has had substance use in his past. We interview him about that, the experiences that he's had, and it's not the most mathematical episode that we've ever done, but it's part of a three-part series. That's correct, that's correct. Now actually about the math, you know,
00:22:57
Speaker
My idea is that anything that can be described has mathematical element to it. So that's my argument. And again, we can get more into the math and the relationship between parts of your brain and consciousness. But more on the episode, we hit on so many things. We talk about hypnosis. We talk about dreams. We talk about, oh, the evolution of creativity and inspiration and what that has to do with consciousness as a whole.
00:23:24
Speaker
We even talk a little bit about Gabriel's trip to Mexico and we are going to talk a little bit right now about, I thought it'd be cool to just talk to you about a mathematical theory of mental illness that treats it as a graph. Gabriel, do you want to introduce them to what a graph means speaking mathematically? Well, let's see. So I think that we'd have to start off with what a normal conscious experience would be, right?
00:23:51
Speaker
Oh, yeah. But I mean, I think we shouldn't choose the concept of just graphs. What do you think? Sure. Yes. So obviously, you know, basic, basic definition of a graph, you know, an X, Y coordinate grid that we all learn in fifth grade or in middle school where you just plot points according to an equation. It's pretty well understood.
00:24:08
Speaker
And that's a popular notion of a graph. There's another word, and it's very confusing, and the guest found it confusing on the episode about a graph, is that a graph is also a set of nodes connected by arrows or lines. So I was wrong. No, you're both right. Okay, that was very polite, John. That's a very polite way of you saying, not quite, Gabe, not quite.
00:24:31
Speaker
But yeah, so there's this theory that mental illness is different than physical illness in the way that it can be treated. Because with mental illness, if you really think about it, you might only have symptoms in certain cases. It's not least controversial thing. But when you are treating symptoms, a lot of times you could treat groups of symptoms. And if you treat them as groups that can be looked at under this graph,
00:25:01
Speaker
You could use certain illnesses based on the shape of the graph. So think about it like the shape of the Internet. If the shape of the Internet represented an illness, then you'd treat it based on if you're on a very link heavy site like Wikipedia, or if you're on a very content heavy site like Huffington Post, not Huffington Post, but something like that.
00:25:23
Speaker
Interesting. Interesting. Wow. Yeah, there's a lot of really cool ways of describing. Oh, and I wanted to say one other thing. I also talk about a personal story of a family member that was diagnosed with severe bipolar disorder and was very successfully treated with electroconvulsive therapy. I'm not going to say a whole lot here, but be sure to tune in next week for that episode.
Conclusion and Future Insights
00:25:47
Speaker
And I reveal something personal about myself. But yeah, so this has been the end of the minisode that wasn't part of the minisode. This page intentionally left blank. Very good. We'll see you guys next time.