WEBVTT Kind: captions; language: en-us NOTE Treffsikkerhet: 91% (H?Y) 00:00:00.000 --> 00:00:09.700 In this video we will talk about some important concepts in the domain of probability, so we will see 00:00:09.700 --> 00:00:21.150 things like random events, independent events, multiple trials, multiple configurations, or paths, and 00:00:21.150 --> 00:00:28.650 repeatable, or predictable patterns and how these are all related with the concept of probability. NOTE Treffsikkerhet: 86% (H?Y) 00:00:28.650 --> 00:00:36.350 To understand this we will go through eight simple and trivial example, which is the coin toss. 00:00:36.350 --> 00:00:42.600 Because this is something we can easily understand and think about, and then we can transfer what we 00:00:42.600 --> 00:00:52.500 learn from this to more realistic situations that are relevant for our study. So pause this videom go 00:00:52.500 --> 00:00:59.700 get a coin, toss it ten times and write down if you get heads or tails each time. NOTE Treffsikkerhet: 86% (H?Y) 00:01:01.200 --> 00:01:10.400 I did this and I got this result, so I got heads the first time, and then heads the second time, tails 00:01:10.400 --> 00:01:12.949 the third time, and so on. NOTE Treffsikkerhet: 90% (H?Y) 00:01:12.949 --> 00:01:21.050 This is an actual sequence of ten coins flipped, well the same coin flip 10 times actually. NOTE Treffsikkerhet: 91% (H?Y) 00:01:21.050 --> 00:01:31.800 Okay so what about that ? Let's do some counts so we can see and calculate what is expected and what 00:01:31.800 --> 00:01:43.050 is actual. In the first 2 tosses I got two heads, so I'm going to write this down as two head, 0 tails. NOTE Treffsikkerhet: 88% (H?Y) 00:01:43.050 --> 00:01:53.700 If you look at my first four, I got one, two, three heads and one tails, and so I can write that as 3 00:01:53.700 --> 00:02:03.800 heads one Tails. If you look at the whole sequence, then in total I got seven heads and three tails. NOTE Treffsikkerhet: 91% (H?Y) 00:02:06.700 --> 00:02:14.500 In past years we would have everyone in class physically do this in the course of the teaching 00:02:14.500 --> 00:02:21.300 statistics and write down what they got and then we would coun.t Unfortunately this is not possible 00:02:21.300 --> 00:02:28.500 now, but what we can do is have the computer do simulations for us that would be like a 00:02:28.500 --> 00:02:30.000 classroom. NOTE Treffsikkerhet: 91% (H?Y) 00:02:30.000 --> 00:02:37.800 But you can still go through the experience with some friends or other students. NOTE Treffsikkerhet: 91% (H?Y) 00:02:38.100 --> 00:02:43.000 So first let's look at the possibilities. NOTE Treffsikkerhet: 91% (H?Y) 00:02:43.700 --> 00:02:52.100 The first time we toss a coin it can be either heads or tails ,and it's equally likely to be either 00:02:52.100 --> 00:02:53.400 one of them. NOTE Treffsikkerhet: 91% (H?Y) 00:02:53.400 --> 00:03:00.500 Then the second time, well if you were heads the first time and you get heads the second time, then 00:03:00.500 --> 00:03:06.200 the pattern is heads heads and the count is twice heads 0 tails. NOTE Treffsikkerhet: 91% (H?Y) 00:03:07.400 --> 00:03:14.600 If you got heads the first time the other possibility is you get tails the second time, so this would 00:03:14.600 --> 00:03:18.400 be your pattern and this would be the count. NOTE Treffsikkerhet: 91% (H?Y) 00:03:18.700 --> 00:03:26.800 If you got tails the first time ,you could get heads the second time and get this pattern and 00:03:26.800 --> 00:03:34.700 this count, one of each, or you could get tails the second time as well so you end up with Tails twice 00:03:34.700 --> 00:03:36.500 and no heads. NOTE Treffsikkerhet: 90% (H?Y) 00:03:36.500 --> 00:03:45.000 So the only way to get heads twice is to get heads in the first and the second toss, but you can get 00:03:45.000 --> 00:03:52.600 heads and tails by either having heads first tails second, or tails first and head second. NOTE Treffsikkerhet: 91% (H?Y) 00:03:53.400 --> 00:04:02.900 The probability of any pattern is how many ways you can get this pattern and how many different 00:04:02.900 --> 00:04:10.800 outcomes are possible, so how many ways divided over how many possibilities there are. Let's apply 00:04:10.800 --> 00:04:13.899 this to our actual situation. NOTE Treffsikkerhet: 91% (H?Y) 00:04:13.899 --> 00:04:21.899 So these are the possibilities we can have this, or this, or this, or this, there are four different 00:04:21.899 --> 00:04:31.350 outcomes that are possible when you toss a coin just twice. And of these four outcomes only one 00:04:31.350 --> 00:04:41.600 includes two heads, one includes two tails, and two have one of each. So how likely are you to get two 00:04:41.600 --> 00:04:43.850 heads ? Well there's only one way to NOTE Treffsikkerhet: 79% (H?Y) 00:04:43.850 --> 00:04:55.200 to do that, out of four possible different outcomes one divided by four is 0.25, which is 25%. NOTE Treffsikkerhet: 84% (H?Y) 00:04:55.900 --> 00:05:03.000 Exactly the same is true for getting two tails there's only one way to get two tails out of four 00:05:03.000 --> 00:05:14.500 possible things that can happen that's 0.25 or 25%. But there are two ways to get one of each and two 00:05:14.500 --> 00:05:19.950 divided by four is 0.5, or 50%, that's 1/2. NOTE Treffsikkerhet: 91% (H?Y) 00:05:19.950 --> 00:05:28.000 So these are the expected probabilities of occurrence for three different kinds of counts, getting 00:05:28.000 --> 00:05:39.600 two heads or two tails, or one of each. What if we toss a third time ? Well by the same kind of thinking, 00:05:39.600 --> 00:05:47.500 after having heads twice we can either get heads or tails. After having heads tails we can get heads 00:05:47.500 --> 00:05:50.250 or tails after having tails and heads NOTE Treffsikkerhet: 80% (H?Y) 00:05:50.250 --> 00:05:58.950 we can get heads or tails and so on, each time we can again get one of the two options with equal 00:05:58.950 --> 00:06:05.600 probability. And these are the possible patterns so three times heads, two times heads one tails and 00:06:05.600 --> 00:06:13.300 so on, and these are the counts. Obviously the only way to get three heads in three tosses, is if you 00:06:13.300 --> 00:06:18.400 get heads in the first, and the second, and the third toss and so on. NOTE Treffsikkerhet: 81% (H?Y) 00:06:18.400 --> 00:06:22.200 What if you do a fourth toss? NOTE Treffsikkerhet: 91% (H?Y) 00:06:22.300 --> 00:06:32.000 Again after each possibility of the third we have two possibilities of equal probability in 00:06:32.000 --> 00:06:38.100 the fourth, and these are the possible patterns, these are all the possible resulting patterns. They're 00:06:38.100 --> 00:06:47.100 all equally likely, because each one comes after a series of events that are equally likely and if 00:06:47.100 --> 00:06:51.800 these are the corresponding counts. And we're not going to go through all of them . NOTE Treffsikkerhet: 83% (H?Y) 00:06:52.100 --> 00:07:01.400 Just to note that each outcome, that each coin toss can lead to two options for the next one that are 00:07:01.400 --> 00:07:09.400 equally likely. So there are all these possible paths one can take and every person who tosses a coin 00:07:09.400 --> 00:07:20.100 four times will necessarily take one of these paths. These are all the possible options. These are 00:07:20.100 --> 00:07:22.799 all the possible patterns of results one NOTE Treffsikkerhet: 91% (H?Y) 00:07:22.799 --> 00:07:27.950 can get and well how many are there ? NOTE Treffsikkerhet: 91% (H?Y) 00:07:27.950 --> 00:07:38.700 Well there's two options, first when you first toss the coin once, and then there's two options when 00:07:38.700 --> 00:07:45.100 you toss the second time, so the total after two tosses is four different things that can happen, but 00:07:45.100 --> 00:07:52.200 we've already seen that. Then you toss again wth two options, so you double the number of possible 00:07:52.200 --> 00:07:57.400 outcomes, and then you toss again with two options, and again you double the number of possible 00:07:57.400 --> 00:07:58.950 outcomes. NOTE Treffsikkerhet: 80% (H?Y) 00:07:58.950 --> 00:08:04.700 So there's 16 different patterns that you can get. NOTE Treffsikkerhet: 91% (H?Y) 00:08:05.800 --> 00:08:09.549 Let's generalize this a little bit. NOTE Treffsikkerhet: 91% (H?Y) 00:08:09.549 --> 00:08:16.500 So at each trial, each time we toss a coin, we multiply by how many possibilities there are, in this 00:08:16.500 --> 00:08:24.600 case there are two. So one toss, there's two possibilities we get heads or tails, two tosses, is two times 00:08:24.600 --> 00:08:32.650 two is four possibilities, three tosses is eight possibilities, four tosses is 16 possibilities. NOTE Treffsikkerhet: 75% (MEDIUM) 00:08:32.650 --> 00:08:41.400 So if you look at these essentially we are multiplying 2 by itself as many times as we toss the coin. NOTE Treffsikkerhet: 91% (H?Y) 00:08:41.400 --> 00:08:49.350 So one toss is two just multiplied by nothing, just two so once. NOTE Treffsikkerhet: 76% (H?Y) 00:08:49.350 --> 00:08:59.500 Two tosses is two multiplied by self two times, three tosses is multiplying two by itself three times, four tosses 00:08:59.500 --> 00:09:02.200 multiplying two by itself four times. NOTE Treffsikkerhet: 78% (H?Y) 00:09:02.200 --> 00:09:11.450 So this is 2 to the 4th power, and in general if you have any number of tosses and n tosses, there are 2 00:09:11.450 --> 00:09:18.200 to the nth power possibilities, two to the end different possible outcomes. NOTE Treffsikkerhet: 91% (H?Y) 00:09:20.200 --> 00:09:26.000 Let's see how many different kinds of outcomes we can get. NOTE Treffsikkerhet: 85% (H?Y) 00:09:26.000 --> 00:09:34.400 So there is only one way to get four heads in four coin tosses. That's because the only way to get 00:09:34.400 --> 00:09:40.400 four heads is to get heads in the first, and the second ,and the third, and the fourth. NOTE Treffsikkerhet: 86% (H?Y) 00:09:40.400 --> 00:09:50.750 How many ways are there to get three heads and once Tails ? Well there is four possible places where 00:09:50.750 --> 00:09:57.700 tails can come up, so they can come up in the first time, or the second time, or the third time, or the 00:09:57.700 --> 00:10:03.600 fourth time. These are the only ways you can get three heads, so there's four of those because 00:10:03.600 --> 00:10:08.350 there are four positions the only tails can be in. NOTE Treffsikkerhet: 84% (H?Y) 00:10:08.350 --> 00:10:13.300 And there are six ways you can get two of each. NOTE Treffsikkerhet: 90% (H?Y) 00:10:13.500 --> 00:10:17.700 We're not going to go through these one by one. NOTE Treffsikkerhet: 79% (H?Y) 00:10:20.400 --> 00:10:31.250 So the probabilities for getting these different outcomes are according to the rule we showed before, NOTE Treffsikkerhet: 78% (H?Y) 00:10:31.250 --> 00:10:40.800 dividing by 16, because there are 16 different patterns, there is only one way to get four heads so 00:10:40.800 --> 00:10:50.800 one out of 16 is 6.25 percent. That's how likely you are to get four heads in a row. NOTE Treffsikkerhet: 87% (H?Y) 00:10:51.100 --> 00:10:59.400 The probability of getting three heads and once tails in four coin tosses, there are four ways to get 00:10:59.400 --> 00:11:06.350 that divided by 16 total possibilities so that's 25%. NOTE Treffsikkerhet: 78% (H?Y) 00:11:06.350 --> 00:11:16.800 The probability of getting 2 of each is 6 / 16 that's 37 and a half percent, and likewise for once 00:11:16.800 --> 00:11:22.800 once heads is the same as three Tails, so it's the same as three heads in terms of probability. 00:11:22.800 --> 00:11:29.400 And no heads is the same as all tails, which from the probabilistic point of view 00:11:29.400 --> 00:11:36.100 is the same as four heads, so it's 1 over 16. You can go back and see the actual patterns and count them 00:11:36.100 --> 00:11:37.050 if you like. NOTE Treffsikkerhet: 91% (H?Y) 00:11:37.050 --> 00:11:41.350 Ao these are the expected probabilities, NOTE Treffsikkerhet: 89% (H?Y) 00:11:41.350 --> 00:11:50.300 and a very important property of these is because they are all the possibilities, we've exhausted all 00:11:50.300 --> 00:11:55.600 the possibilities of what can happen, there's nothing else that's possible if you toss a coin four 00:11:55.600 --> 00:12:03.400 times you can get either four heads, or three heads, or two, or one, or none. That's it, so there's 16 00:12:03.400 --> 00:12:10.800 things that can happen, and these are all of them. So if you add 1, plus 4, plus 6, plus 4, plus 1, you get 00:12:10.800 --> 00:12:12.050 16. NOTE Treffsikkerhet: 82% (H?Y) 00:12:12.050 --> 00:12:19.700 And if you add the corresponding probabilities, they must add to a hundred percent. So these are the 00:12:19.700 --> 00:12:26.750 probabilitie.s they add up to one. So the total probability is 1, it has to be one. What does it mean 00:12:26.750 --> 00:12:34.400 one, 1 probability of 1 means certain. What are we certain about? We are certain that something will 00:12:34.400 --> 00:12:40.900 happen. I know that doesn't sound like much but this is what it means. we are certain that one of 00:12:40.900 --> 00:12:41.550 these NOTE Treffsikkerhet: 83% (H?Y) 00:12:41.550 --> 00:12:50.700 patterns will occur. So the sum of the probabilities of all possible events adds up to one, because we 00:12:50.700 --> 00:12:56.900 are a hundred percent sure that one of these will happen. Of course we don't know which one, but we 00:12:56.900 --> 00:13:00.100 know how likely each one of them is. NOTE Treffsikkerhet: 82% (H?Y) 00:13:00.100 --> 00:13:11.099 Now if you recall we actually tossed the coin 10 times, what about the possible outcomes from 10 tosses. 00:13:11.099 --> 00:13:17.900 Don't panic we're not going to start counting, but we can still calculate some things easily. NOTE Treffsikkerhet: 82% (H?Y) 00:13:17.900 --> 00:13:25.100 So how many different possible outcomes are there ? Well it's 2 multiplied by itself 10 times, and if you 00:13:25.100 --> 00:13:32.350 bothered to do that you'll see that it is just over a thousand, it's 1024 different possible outcomes. 00:13:32.350 --> 00:13:42.100 Okay so how likely is it to get 10 heads in 10 coin tosses, well there's obviously only one way to 00:13:42.100 --> 00:13:48.200 get that, you have to get heads every time for this to happen. So there's NOTE Treffsikkerhet: 89% (H?Y) 00:13:48.200 --> 00:14:00.400 one way for this to happen divided by 1024 possible pattern, and this number is about one per 00:14:00.400 --> 00:14:05.250 thousand. Just a little bit less than one per thousand NOTE Treffsikkerhet: 82% (H?Y) 00:14:05.250 --> 00:14:15.050 or 0.1% . This looks like a very small number and indeed you would not expect to get 10 heads in a row. 00:14:15.050 --> 00:14:18.400 You would probably be very surprised if this happened. NOTE Treffsikkerhet: 91% (H?Y) 00:14:18.400 --> 00:14:22.050 And you should be, however NOTE Treffsikkerhet: 77% (H?Y) 00:14:22.050 --> 00:14:30.400 if we ask a few thousand people and they all agreed to toss a coin 10 times each NOTE Treffsikkerhet: 91% (H?Y) 00:14:30.600 --> 00:14:40.600 what would you expect, well if nobody got ten heads in a row you should be surprised. Of course the 00:14:40.600 --> 00:14:49.200 person or persons who do get 10 heads in a row will be very surprised indeed, but if they don't occur 00:14:49.200 --> 00:14:56.050 then you knowing about the thousands of people, you should be surprised instead. NOTE Treffsikkerhet: 91% (H?Y) 00:14:56.050 --> 00:15:07.900 So this means that an event that in itself is unlikely, less than one per thousand to happen, becomes 00:15:07.900 --> 00:15:15.500 expected if you try enough times if you have enough people tossing coins. NOTE Treffsikkerhet: 87% (H?Y) 00:15:16.700 --> 00:15:27.500 Okay what have we heard so far, what are the important lessons from here ? So one important thing to 00:15:27.500 --> 00:15:37.100 always keep in mind is to think of probability as frequency, as how often something happens, how often 00:15:37.100 --> 00:15:43.600 we can expect something to happen. So probability is also a kind of expectation, something that's high 00:15:43.600 --> 00:15:46.300 probability is expected, something that's low NOTE Treffsikkerhet: 90% (H?Y) 00:15:46.300 --> 00:15:50.400 probability is unexpected or surprising. NOTE Treffsikkerhet: 88% (H?Y) 00:15:51.000 --> 00:15:58.400 The second important thing to remember is that the total probability, when we have exhausted all the 00:15:58.400 --> 00:16:07.200 possibilities of outcomes, has to be one, 100%,this means something will happen. NOTE Treffsikkerhet: 91% (H?Y) 00:16:08.900 --> 00:16:19.600 And the probability of an outcome depends on the trials, it depends on how many times we try, how many 00:16:19.600 --> 00:16:28.800 people try it, or in how many different ways it can occur. Number of tries or number of different ways, 00:16:28.800 --> 00:16:39.200 and so low probability events will happen eventually if you try enough times or if enough NOTE Treffsikkerhet: 47% (MEDIUM) 00:16:39.200 --> 00:16:40.900 people try. NOTE Treffsikkerhet: 91% (H?Y) 00:16:42.400 --> 00:16:51.050 A random event is an unpredictable event, indeed that's what random means, it means you cannot predict 00:16:51.050 --> 00:17:01.700 what will happen. However if you have lots of independent random events, then you can predict what 00:17:01.700 --> 00:17:09.700 will happen collectively, even though you cannot predict what will happen with each one of them. So 00:17:09.700 --> 00:17:13.250 these are the most important lessons that you should keep in mind NOTE Treffsikkerhet: 91% (H?Y) 00:17:13.250 --> 00:17:19.099 as we go on to discuss more about probability later in this course.