1 00:00:00,000 --> 00:00:15,980 Hello everyone, welcome to the Magnificence of Mathematics. I'm your host, Eddie Kingston. 2 00:00:15,980 --> 00:00:20,760 Today we'll be talking about my favorite subfield of math, probability. You're most 3 00:00:20,760 --> 00:00:25,940 likely already intuitively familiar with probability, say when you're flipping a coin. You know 4 00:00:25,940 --> 00:00:31,720 that there's at least roughly a 50% chance of the coin landing on heads, and a 50% chance 5 00:00:31,720 --> 00:00:36,720 of the coin landing on tails. But here's a head scratcher. Is it true that events that 6 00:00:36,720 --> 00:00:42,920 have a 0% probability of happening can still happen? Obviously not, right? Anything with 7 00:00:42,920 --> 00:00:47,880 a 0% chance of happening can't happen. You can't randomly choose dragon in a game of 8 00:00:47,880 --> 00:00:51,880 rock paper scissors. Well, it might surprise you that the answer to the question I just 9 00:00:51,880 --> 00:00:58,040 posed is in fact, yes. Let's build up to this. Suppose you randomly pick a number either 10 00:00:58,040 --> 00:01:04,720 1 or 2 with equal probability. That is, the probability of picking the number 1 is 1 half, 11 00:01:04,720 --> 00:01:09,360 and the probability of picking the number 2 is 1 half. Now suppose you randomly pick 12 00:01:09,360 --> 00:01:15,600 a number either 1, 2, or 3 with equal probability. Then the probability of picking the number 13 00:01:15,600 --> 00:01:23,080 1, or 2, or 3 is 1 third, which is smaller than 1 half. If you randomly pick a number 14 00:01:23,080 --> 00:01:30,160 1, 2, 3, or 4 with equal probability, the probability of picking the number 1 is 1 over 15 00:01:30,160 --> 00:01:34,960 4, which again is smaller than 1 third and 1 half. At this point, you might have noticed 16 00:01:34,960 --> 00:01:40,960 two things. First, the probability of picking the number 1 from the numbers 1, 2, 3, all 17 00:01:40,960 --> 00:01:46,480 the way up to some positive whole number n is 1 over n, since there are n numbers to 18 00:01:46,480 --> 00:01:53,600 choose from. Second, this probability gets smaller as n gets larger. Now, what if you 19 00:01:53,600 --> 00:01:58,680 wanted to choose a number from the set of all positive integers 1, 2, all the way up 20 00:01:58,680 --> 00:02:04,080 to infinity? Since there are an infinite amount of numbers to choose from, the probability 21 00:02:04,080 --> 00:02:09,840 of choosing the number 1, or any number for that matter, is 1 over infinity, which is 22 00:02:09,840 --> 00:02:16,160 just 0. Wait, huh? How could it be 0 when we're still choosing a number? Well, if the 23 00:02:16,160 --> 00:02:21,240 probability in question had been anything other than 0, there would be some number n 24 00:02:21,240 --> 00:02:25,760 that describes the amount of numbers we're choosing from. But we just said that we're 25 00:02:25,760 --> 00:02:31,940 choosing from the infinite list of positive integers, so this can't be the case. Therefore, 26 00:02:31,940 --> 00:02:38,080 we literally have a 0% chance of choosing any positive integer whatsoever, but we still 27 00:02:38,080 --> 00:02:45,040 do choose a number. Argument I just gave is an albeit hand-wavy proof by contradiction, 28 00:02:45,040 --> 00:02:49,440 and the underlying idea behind all of this is the idea of a limit, which is one of the 29 00:02:49,440 --> 00:02:55,040 fundamental building blocks of calculus and beyond. In a similar vein, it's also true 30 00:02:55,040 --> 00:03:01,000 that events with 100% probability don't necessarily happen. Suppose you want to avoid choosing 31 00:03:01,000 --> 00:03:07,160 the number 1 from the numbers 1, 2, and 3. The probability of avoiding the number 1 is 32 00:03:07,160 --> 00:03:13,480 2 thirds, because you have two favorable outcomes out of three possible outcomes. If you want 33 00:03:13,480 --> 00:03:19,720 to avoid choosing the number 1 from 1, 2, 3, and 4, that probability is 3 fourths, which 34 00:03:19,720 --> 00:03:25,320 is greater than 2 thirds, because you have three favorable outcomes out of four possible 35 00:03:25,320 --> 00:03:31,360 outcomes. Then for any positive integer n, the probability of avoiding the number 1 from 36 00:03:31,360 --> 00:03:39,440 the numbers 1, 2, up to n is n minus 1 divided by n, which gets larger as n gets larger. 37 00:03:39,440 --> 00:03:44,120 You might notice that this probability gets closer and closer to 1, but never actually 38 00:03:44,120 --> 00:03:49,200 reaches it unless you have the infinite set of all positive integers, and want to avoid 39 00:03:49,200 --> 00:03:54,920 choosing the number 1. So the probability of avoiding the number 1, or any number for 40 00:03:54,920 --> 00:04:01,280 that matter, is 1, but we still don't avoid picking a number altogether. How cool is that? 41 00:04:01,280 --> 00:04:05,400 Want another counterintuitive way probability works? You might have heard of the famous 42 00:04:05,400 --> 00:04:10,320 Monty Hall problem inspired by Let's Make a Deal host Monty Hall, who ran the show from 43 00:04:10,320 --> 00:04:16,720 1963 until 1987. If you're unfamiliar with the show or the problem itself, imagine you're 44 00:04:16,720 --> 00:04:21,780 put in front of three doors. The host Monty Hall tells you that behind one of the doors 45 00:04:21,780 --> 00:04:27,600 is a brand new car. Behind the other two doors are goats. Your task is to pick the door the 46 00:04:27,600 --> 00:04:34,120 car is hiding behind. Let's say you pick door 1. Monty then opens door 3, revealing a goat. 47 00:04:34,120 --> 00:04:38,520 Now Monty gives you a choice. Do you want to stick with door 1, or change your door 48 00:04:38,520 --> 00:04:43,880 to door 2? The problem is, given all this information, would it be in your best interest 49 00:04:43,880 --> 00:04:49,200 to switch, or to stick with your original choice? Your knee-jerk reaction might be that 50 00:04:49,200 --> 00:04:53,400 it doesn't matter whether you stick with your first choice or switch, since now there's 51 00:04:53,400 --> 00:04:58,000 only two doors the car could be behind and it's a 50-50 chance. But it turns out that 52 00:04:58,000 --> 00:05:03,920 that's wrong. There is a 2 in 3 chance that switching will net you the car. To explain 53 00:05:03,920 --> 00:05:07,760 this, it might be easier to think about the game as if there were 100 doors to choose 54 00:05:07,760 --> 00:05:14,480 from, 99 goats and 1 car. If you pick door 36, and Monty opens every door except door 55 00:05:14,480 --> 00:05:19,760 36 and door 67, at this point you might be inclined to think, yeah, I'm switching. There's 56 00:05:19,760 --> 00:05:24,420 only a 1 in 100 chance I was right to begin with. I probably guessed wrong, the car is 57 00:05:24,420 --> 00:05:29,920 actually behind door 67. And you're right. Given the other 98 doors that have opened, 58 00:05:29,920 --> 00:05:36,040 the probability that the door you first chose was right all along is only 1 out of 100, 59 00:05:36,040 --> 00:05:41,800 meaning the probability that door 67 was the right door is 99 out of 100. If we scale this 60 00:05:41,800 --> 00:05:46,720 back to the original problem, the probability that the door you first chose was right all 61 00:05:46,720 --> 00:05:51,760 along is 1 out of 3, meaning the probability that the car is behind the other unopened 62 00:05:51,760 --> 00:05:58,040 door is 2 out of 3. If you think about randomly choosing 1 door out of 3 or 100 or however 63 00:05:58,040 --> 00:06:03,200 many as an experiment, this paradox became a thing because when thinking about probabilities 64 00:06:03,200 --> 00:06:07,560 and what would be better to do, people thought about the experiment as if the start of the 65 00:06:07,560 --> 00:06:12,520 game didn't happen. They thought of the experiment as having started at the point where they 66 00:06:12,520 --> 00:06:16,960 have to choose to either stick with their door or change it, which is where they get 67 00:06:16,960 --> 00:06:21,880 50% from. But if you think about the experiment as starting when you first choose a door, 68 00:06:21,880 --> 00:06:27,860 then you only have a 1 in 3 or 1 in 100 or 1 in however many chance of choosing the right 69 00:06:27,860 --> 00:06:32,880 door on the first try. This is all well and good, but where did any of this come from? 70 00:06:32,880 --> 00:06:37,880 Why study probability in the first place? Probability came from the desire to quantify 71 00:06:37,880 --> 00:06:44,480 uncertainty, in particular when it comes to gambling. In the mid 1600s, Gerolamo Cardano 72 00:06:44,480 --> 00:06:49,360 showed the use of defining odds as the ratio of the number of favorable outcomes to the 73 00:06:49,360 --> 00:06:55,160 number of unfavorable outcomes. So for example, 1 to 1 odds means there is 1 favorable outcome 74 00:06:55,160 --> 00:06:59,960 and 1 unfavorable outcome, and around the same time, French mathematicians Pierre de 75 00:06:59,960 --> 00:07:05,040 Fermat and Blaise Pascal wrote to each other and laid the foundations of probability as 76 00:07:05,040 --> 00:07:10,440 we know it. In particular, Blaise Pascal is famous for what's known as Pascal's Triangle, 77 00:07:10,440 --> 00:07:14,680 which tells you how many ways you can pick a certain number of items from a larger set 78 00:07:14,680 --> 00:07:19,040 of items. The reason it's called a triangle is because you can write out these numbers 79 00:07:19,040 --> 00:07:25,320 as if you were building a side of a pyramid, i.e. a triangle from the top down. For example, 80 00:07:25,320 --> 00:07:31,120 the top of the triangle, called row 0, is just 1 because there is only one way to pick 81 00:07:31,120 --> 00:07:38,600 0 items from 0 items. That's just doing nothing. Similarly, the row below, row 1, has a 1 on 82 00:07:38,600 --> 00:07:43,320 the left and a 1 right next to it, because again, there's only one way to pick 0 items 83 00:07:43,320 --> 00:07:48,400 from a set of 1 item, which is to just do nothing, and one way to pick 1 item from that 84 00:07:48,400 --> 00:07:53,960 set, which is just picking that item. Now for row 2, there's a 1 on either end because 85 00:07:53,960 --> 00:07:58,960 there's only one way to pick 0 items from a set of 2, but not doing anything, and only 86 00:07:58,960 --> 00:08:05,200 one way to pick 2 items from a set of 2 by picking both items, and a 2 in the middle 87 00:08:05,200 --> 00:08:10,080 because there's 2 ways to pick 1 item from a set of 2, picking one thing or the other. 88 00:08:10,080 --> 00:08:15,200 For row 3, there's once again a 1 on either end, one way to pick nothing and one way to 89 00:08:15,200 --> 00:08:21,200 pick everything, and 2 3s in the middle. 3 ways to pick 1 object, and 3 ways to pick 90 00:08:21,200 --> 00:08:28,200 2 objects. That is, 3 ways to leave 1 object out. Let's recap. For row 0, we have a 1. 91 00:08:28,200 --> 00:08:34,480 For row 1, we have a 1 and a 1, which would sandwich the 1 from row 0 if row 0 was brought 92 00:08:34,480 --> 00:08:41,320 down. For row 2, we have a 1, 2, and 1, with the 2 sandwiched between the 2 1s from row 93 00:08:41,320 --> 00:08:48,560 1. Row 3 reads 1 3 3 1, and so on. If you write these out, or just look up the triangle, 94 00:08:48,560 --> 00:08:53,080 you might notice that if you take any two adjacent numbers in a certain row, and look 95 00:08:53,080 --> 00:08:57,400 at the number in the row below that's sandwiched between those two numbers, the sandwiched 96 00:08:57,400 --> 00:09:01,680 number is the sum of the two numbers directly above it. So if you look at the left side 97 00:09:01,680 --> 00:09:09,440 of row 2 and row 3, you see a 1 and 2 in row 2, a 1 on the far left in row 3, and a 3 between 98 00:09:09,440 --> 00:09:15,120 the 1 and the 2 in row 2, and 1 plus 2 is 3. Another way these are read is in the choose 99 00:09:15,120 --> 00:09:21,760 notation. So row 2 can be read as 2 choose 0, which equals 1, 2 choose 1, which equals 100 00:09:21,760 --> 00:09:28,480 2, and 2 choose 2, which equals 1. 2 choose 1, for example, can be interpreted as, from 101 00:09:28,480 --> 00:09:33,480 a set of two objects, choose one of them. This triangle is crucial for beginning to 102 00:09:33,480 --> 00:09:38,520 understand the field of combinatorics, which is what mathematicians fancily call counting. 103 00:09:38,520 --> 00:09:42,760 If you were to take a beginning college level probability class, you'd likely encounter 104 00:09:42,760 --> 00:09:47,120 basic combinatorics at the beginning of the class, and be able to answer questions like, 105 00:09:47,120 --> 00:09:52,840 if you have a hand of 5 standard playing cards, what's the probability of having 2 aces? To 106 00:09:52,840 --> 00:09:58,200 answer this, you want 2 of the 4 aces from the deck in your hand, so there's 4 choose 107 00:09:58,200 --> 00:10:04,560 2, which is 6 ways to choose the aces for your hand. Now you don't want any more aces, 108 00:10:04,560 --> 00:10:11,040 so you have to pick 3 non-ace cards out of the remaining 48 possible cards. So you have 109 00:10:11,040 --> 00:10:18,080 48 choose 3, which equals 17,296 ways to pick the remaining cards. Multiplying these together 110 00:10:18,080 --> 00:10:25,560 means you have 103,776 favorable outcomes. Since you have 52 cards, and you choose 5 111 00:10:25,560 --> 00:10:35,560 of them for your hand, you have 52 choose 5, which equals 2,598,960 total possible outcomes. 112 00:10:35,560 --> 00:10:40,180 So the answer we're looking for is the number of favorable outcomes divided by the number 113 00:10:40,180 --> 00:10:51,240 of possible outcomes, giving us 103,776 divided by 2,598,960, which comes out to be slightly 114 00:10:51,240 --> 00:10:56,080 less than 4%. If you've ever played poker or powerball or something, this is how the 115 00:10:56,080 --> 00:11:01,040 probability of certain hands or certain amounts of numbers matched are calculated. If I haven't 116 00:11:01,040 --> 00:11:05,120 broken your brain by this point, I want to talk a bit about how mathematicians think 117 00:11:05,120 --> 00:11:10,160 of probability. Two of the famous mathematicians to get the ball rolling are André Marcot 118 00:11:10,160 --> 00:11:16,200 and André Kolmogorov in the early 1900s. Marcot was largely responsible for the development 119 00:11:16,200 --> 00:11:21,000 of stochastic processes, which are sequences of random events that happen over time. Going 120 00:11:21,000 --> 00:11:25,800 back to our coin flipping example, suppose you play a game where you win $1 every time 121 00:11:25,800 --> 00:11:31,320 the coin lands on heads, and you lose $1 every time the coin lands on tails. The amount of 122 00:11:31,320 --> 00:11:36,560 money you win or lose is a stochastic process. You can map out how the game goes over time, 123 00:11:36,560 --> 00:11:40,320 and you'll get a different map every time you play the game, assuming you play enough 124 00:11:40,320 --> 00:11:45,040 of course. Stochastic processes can be used to model all kinds of things like wait times 125 00:11:45,040 --> 00:11:50,120 and even stock prices. Kolmogorov established the foundations of modern probability via 126 00:11:50,120 --> 00:11:55,320 measure theory. What is measure theory, you may ask? Well, a measure is how mathematicians 127 00:11:55,320 --> 00:12:00,120 generalize the notions of measuring things as you know them, like areas, volumes, and 128 00:12:00,120 --> 00:12:05,360 masses. To define a measure, a measure space, and a probability space, we need three key 129 00:12:05,360 --> 00:12:10,440 ingredients. The first ingredient is a set, which is a collection of distinct objects, 130 00:12:10,440 --> 00:12:16,480 say numbers. So let's just say we have the set containing the numbers 1, 2, and 3. These 131 00:12:16,480 --> 00:12:22,700 are our elements in our set. Let's call this set X. The second ingredient is what's known 132 00:12:22,700 --> 00:12:29,220 as a sigma algebra. This is a collection of subsets of our original set, meaning a collection 133 00:12:29,220 --> 00:12:35,600 of sets that comprise either part of or all of our original set. This sigma algebra satisfies 134 00:12:35,600 --> 00:12:41,640 three important properties. The first is that the entire set X is an element of our sigma 135 00:12:41,640 --> 00:12:47,620 algebra. The second property is that the sigma algebra is closed under complements, meaning 136 00:12:47,620 --> 00:12:54,240 if any subset of X is in our sigma algebra, then so is every element in X outside of that 137 00:12:54,240 --> 00:12:59,200 subset. For example, if the set containing just the number 1 is in our sigma algebra, 138 00:12:59,200 --> 00:13:03,440 then so is the set containing the numbers 2 and 3. An immediate consequence of this 139 00:13:03,440 --> 00:13:08,160 second property is that the set containing nothing, or the so-called empty set, is in 140 00:13:08,160 --> 00:13:13,320 our sigma algebra. Finally, the third property is that if you have a countable collection 141 00:13:13,320 --> 00:13:18,520 of subsets in our sigma algebra, meaning you can assign each subset a number, then the 142 00:13:18,520 --> 00:13:23,840 union of those subsets is also in the sigma algebra. This just means you're able to combine 143 00:13:23,840 --> 00:13:27,800 elements from two or more distinct subsets into one big subset. 144 00:13:27,800 --> 00:13:33,200 So in our example, since we said the set containing 1 and the set containing 2 and 3 are both 145 00:13:33,200 --> 00:13:39,040 in the sigma algebra, the set containing 1, 2, and 3 is in the sigma algebra, but we do 146 00:13:39,040 --> 00:13:43,360 that anyway since that's just the original set X, which we established was in the sigma 147 00:13:43,360 --> 00:13:45,400 algebra via the first property. 148 00:13:45,400 --> 00:13:51,360 Okay, let's take a step back. So far we have two ingredients, a set X and a collection 149 00:13:51,360 --> 00:13:57,060 of subsets of X, called a sigma algebra, satisfying three different properties. The third and 150 00:13:57,060 --> 00:14:03,000 final ingredient is what's known as a measure. A measure is a function that takes in a subset 151 00:14:03,000 --> 00:14:09,240 of X from our sigma algebra and outputs either a real number, negative infinity, or positive 152 00:14:09,240 --> 00:14:15,160 infinity. You can assign values to these sigma algebra elements in any way as long as it 153 00:14:15,160 --> 00:14:20,400 obeys three rules. First, the measure of any subset has to be greater than or equal to 154 00:14:20,400 --> 00:14:27,040 zero. Second, the measure of the empty set is zero. Third, the measure of the union of 155 00:14:27,040 --> 00:14:32,600 disjoint subsets, meaning subsets that don't share any elements, is equal to the sum of 156 00:14:32,600 --> 00:14:38,360 the measures of the disjoint subsets. So say you have two subsets, the intervals zero to 157 00:14:38,360 --> 00:14:43,240 one inclusive and the interval one to two inclusive. If you picture these two intervals 158 00:14:43,240 --> 00:14:48,400 on a number line, you might say the distance between zero and one is one and the distance 159 00:14:48,400 --> 00:14:53,440 between one and two is one. So the distance between zero and two is just one plus one 160 00:14:53,440 --> 00:14:58,680 equals two. So these three ingredients, a set, a sigma algebra, and a measure, come 161 00:14:58,680 --> 00:15:03,420 together to create what's known as a measure space. This general notion of a measure space 162 00:15:03,420 --> 00:15:08,040 is important in analysis, which you can think of as an extension and or generalization of 163 00:15:08,040 --> 00:15:13,880 calculus. A probability space is just a measure space where the measure of the whole set X 164 00:15:13,880 --> 00:15:19,560 is equal to one. We can think of probabilities as measures and the set of all possible outcomes 165 00:15:19,560 --> 00:15:25,840 of an experiment as our set X. For example, flipping a coin once has two possibilities, 166 00:15:25,840 --> 00:15:30,360 heads or tails, and we can ignore the possibility of the coin landing on its side and say the 167 00:15:30,360 --> 00:15:36,320 probability of getting either heads or tails is one or 100%. The assignment of probability 168 00:15:36,320 --> 00:15:41,000 to certain events are usually determined by distributions, think bell curves if you've 169 00:15:41,000 --> 00:15:46,260 ever seen those. I'll talk more about distributions next time. Komagoro was able to use this framework 170 00:15:46,260 --> 00:15:50,640 to establish the foundations of all modern probability, but to talk about it much further 171 00:15:50,640 --> 00:15:55,280 requires ideas from calculus, which I'll talk about in upcoming episodes. Thank you 172 00:15:55,280 --> 00:15:59,840 for tuning into this episode of the Magnificence of Mathematics. In the next episode or two, 173 00:15:59,840 --> 00:16:04,040 I'll talk more about statistics, my time in grad school, and advice for undergrads 174 00:16:04,040 --> 00:16:20,720 and grad students in general. Hope to see you then. 175 00:16:20,720 -->00:17:28,080 Algid Productions LLC Outro. Thank you for listening!