1 00:00:00,090 --> 00:00:03,420 We've been starting to look at how to answer probability questions. 2 00:00:03,420 --> 00:00:08,790 And at this point we want to make sure that we understand how to calculate probability for discrete 3 00:00:08,790 --> 00:00:09,910 random variables. 4 00:00:09,930 --> 00:00:14,970 Now, we want to distinguish here between discrete and continuous random variables. 5 00:00:14,970 --> 00:00:22,410 So continuous random variables are variables representing data sets that we can think of as being continuous. 6 00:00:22,410 --> 00:00:29,640 So think about, for instance, dimensions like length, width and height, or maybe even area or volume. 7 00:00:29,670 --> 00:00:36,330 One common example is the height of people we would call the height of people a continuous random variable, 8 00:00:36,360 --> 00:00:42,480 because in theory people can take on any height, at least maybe within a particular range. 9 00:00:42,480 --> 00:00:49,050 So we might say that somebody measures five feet six inches tall and then another person measures five 10 00:00:49,050 --> 00:00:51,570 feet seven inches tall. 11 00:00:51,600 --> 00:00:58,230 But we also recognize that there are many people who have some height in between five, six and five 12 00:00:58,230 --> 00:00:58,680 seven. 13 00:00:58,680 --> 00:01:03,720 We might have somebody who measures exactly five, six and a half or somebody who measures exactly five, 14 00:01:03,720 --> 00:01:04,709 six and a quarter. 15 00:01:04,739 --> 00:01:10,920 And if we go even more granular in our measurement, in theory, we could measure an infinite number 16 00:01:10,920 --> 00:01:13,560 of different heights between five, six and five seven. 17 00:01:13,590 --> 00:01:18,030 The only thing that's holding us back here is how granular we can get with our measurement. 18 00:01:18,030 --> 00:01:24,240 But the concept is that if we had the ability to measure with more and more and more accurate precision, 19 00:01:24,240 --> 00:01:30,480 we could find many different heights and infinite number of heights between five, six and five seven. 20 00:01:30,480 --> 00:01:34,700 And so we can define the height of people as a continuous random variable. 21 00:01:34,710 --> 00:01:38,070 The same would be true for time in normal life. 22 00:01:38,070 --> 00:01:42,990 We might only be interested in measuring time in terms of maybe seconds, minutes or hours. 23 00:01:42,990 --> 00:01:49,620 But at the Olympics, when we're talking about athletes who race down to the 10th of a second or 100th 24 00:01:49,620 --> 00:01:54,840 of a second or even thousandth of a second, as long as we could continue measuring at a more and more 25 00:01:54,840 --> 00:01:59,550 granular level, we can continue to distinguish between lengths of time. 26 00:01:59,550 --> 00:02:02,460 And so time would be considered a continuous random variable. 27 00:02:02,490 --> 00:02:06,090 Contrast this with the idea of discrete random variables. 28 00:02:06,090 --> 00:02:09,720 The key here we want to think about is things that are countable. 29 00:02:09,720 --> 00:02:12,960 For instance, number of children in a family. 30 00:02:12,960 --> 00:02:18,450 It's impossible for a family to have 2.7 children or 4.2 children. 31 00:02:18,450 --> 00:02:25,650 That family is always going to have zero one, two, three, etc. Number of children. 32 00:02:25,650 --> 00:02:28,320 The number of children in the family are countable. 33 00:02:28,350 --> 00:02:32,430 Or we could think about something like flipping a coin. 34 00:02:32,430 --> 00:02:39,660 So coin flips and the number of heads that we get in those coin flips is always going to be zero one, 35 00:02:39,660 --> 00:02:47,820 two, three, etc. It's impossible if we flip a coin two times to get half of a head or one and a half 36 00:02:47,820 --> 00:02:54,780 heads, we're always going to get zero one or two heads in two coin flips and there's no in between. 37 00:02:54,780 --> 00:03:00,330 Even if we could get more granular with our measurement, it doesn't make sense to say that we get one 38 00:03:00,330 --> 00:03:02,700 and one half heads on two coin flips. 39 00:03:02,700 --> 00:03:06,780 There's only these discrete values and nothing in between those values. 40 00:03:06,780 --> 00:03:13,110 So just think about discrete random variables as countable variables with no logical in between values. 41 00:03:13,110 --> 00:03:19,650 Whereas with continuous random variables we have in theory an infinite number of in between values. 42 00:03:19,650 --> 00:03:25,170 And as long as we can get specific enough in our measurements, we could continue finding more and more 43 00:03:25,170 --> 00:03:26,730 in between values. 44 00:03:26,730 --> 00:03:32,340 So with that in mind, what we want to do now is make sure we understand how to do probability calculations 45 00:03:32,340 --> 00:03:35,400 when we have a discrete random variable. 46 00:03:35,700 --> 00:03:42,090 So let's go back to this idea of the coin flip and let's say that we flip a coin one time. 47 00:03:42,390 --> 00:03:44,370 There are only two possible outcomes. 48 00:03:44,370 --> 00:03:49,530 When we flip the coin one time, we can either flip heads or we can flip tails. 49 00:03:49,530 --> 00:03:55,530 If we think about that in terms of probability, we could build ourselves a little table here and we 50 00:03:55,530 --> 00:04:00,600 could say that the discrete random variable modeling, the number of heads that we get when we flip 51 00:04:00,600 --> 00:04:02,250 that coin is x. 52 00:04:02,250 --> 00:04:06,600 So this variable X here models the number of heads we get when we flip a coin. 53 00:04:06,600 --> 00:04:11,340 And if we flip the coin one time, there are only two possible outcomes. 54 00:04:11,340 --> 00:04:16,440 One is we flip heads zero times, The other is we flip heads one time. 55 00:04:16,440 --> 00:04:21,899 Since we're only flipping the coin one time, there's no way to flip heads twice or three times. 56 00:04:21,899 --> 00:04:24,360 And obviously negative numbers don't apply here. 57 00:04:24,360 --> 00:04:25,650 So only two outcomes. 58 00:04:25,650 --> 00:04:31,440 We flip heads zero times or we flip heads one time and then we can think about the probability of each 59 00:04:31,440 --> 00:04:34,620 of these outcomes for this discrete random variable X. 60 00:04:34,620 --> 00:04:37,860 Well, we would indicate that as the probability of X. 61 00:04:37,860 --> 00:04:44,910 And so this value here in our table is the probability that we flip heads zero times when we flip the 62 00:04:44,910 --> 00:04:46,230 coin one time. 63 00:04:46,230 --> 00:04:52,560 Well, the probability of flipping heads zero times when we flip a coin once is equivalent to the probability 64 00:04:52,560 --> 00:04:54,630 of flipping tails on that coin flip. 65 00:04:54,630 --> 00:04:59,820 So that probability is one half the probability that we flip heads once when we flip the. 66 00:05:00,260 --> 00:05:03,220 One time is also, of course, one half. 67 00:05:03,230 --> 00:05:08,960 So we have a little table here with our discrete random variable x and the probability that we find 68 00:05:08,960 --> 00:05:11,190 this particular value of x. 69 00:05:11,210 --> 00:05:17,300 Now, if we want to represent that in a probability distribution, here's what that looks like. 70 00:05:17,300 --> 00:05:19,820 We could make a probability distribution here. 71 00:05:19,820 --> 00:05:26,210 We'll have the values that X can take on along the horizontal axis and along the vertical axis. 72 00:05:26,210 --> 00:05:30,620 We'll have the probability that that particular value of x occurs. 73 00:05:30,620 --> 00:05:32,960 So here's what the distribution would look like. 74 00:05:32,960 --> 00:05:34,700 For one coin flip. 75 00:05:35,000 --> 00:05:41,630 It's this distribution Here we see along the horizontal axis that in one coin flip x, the number of 76 00:05:41,630 --> 00:05:45,740 heads that we get when we flip a coin one time can only take on the values. 77 00:05:45,740 --> 00:05:51,680 Zero or one with one coin flip is only possible to flip heads zero times or one time. 78 00:05:51,680 --> 00:05:54,710 So these are the only values we have along our horizontal axis. 79 00:05:54,710 --> 00:06:02,330 We've put our vertical axis in units of sixteenths so that we can expand on this graph as we add more 80 00:06:02,330 --> 00:06:03,920 flips to our table here. 81 00:06:03,920 --> 00:06:10,460 So if we think about each unit here along the vertical axis as being out of 16, then eight divided 82 00:06:10,460 --> 00:06:12,860 by 16 is the same as one half. 83 00:06:12,860 --> 00:06:17,120 So our probability right here is one half. 84 00:06:17,120 --> 00:06:25,370 Our probability at four over 16 is one fourth, our probability at two over 16 is one eighth. 85 00:06:25,400 --> 00:06:31,100 So since we know from our little probability distribution table here that the probability of flipping 86 00:06:31,100 --> 00:06:38,540 zero heads is one half, we've indicated that here with this bar that extends up to eight over 16 or 87 00:06:38,540 --> 00:06:39,290 one half. 88 00:06:39,290 --> 00:06:45,260 And then same thing here, the probability of flipping heads one time is also one half. 89 00:06:45,260 --> 00:06:48,260 So we have that represented with this bar here. 90 00:06:48,290 --> 00:06:54,830 This is what probability looks like for a single coin flip with the discrete random variable x representing 91 00:06:54,830 --> 00:06:57,800 the number of heads that we flip on that coin flip. 92 00:06:57,830 --> 00:07:05,720 If we now add a second coin flip to this scenario, now we're going to say that we flip a coin two times. 93 00:07:05,720 --> 00:07:09,980 Well, we have to now account for all of these possibilities. 94 00:07:09,980 --> 00:07:15,710 Notice what we did here is that we started with our original single coin flip, and we said that on 95 00:07:15,710 --> 00:07:18,350 our first coin flip, we're either going to get heads or tails. 96 00:07:18,350 --> 00:07:24,380 Now, on our second coin flip, we could get heads after flipping heads on the first flip, or we could 97 00:07:24,380 --> 00:07:26,960 get heads after flipping tails on the first flip. 98 00:07:27,110 --> 00:07:32,690 But we can also have the scenario where we flipped heads and tails on the first flip, but then tails 99 00:07:32,690 --> 00:07:34,070 on the second flip. 100 00:07:34,070 --> 00:07:40,790 So what was two possible outcomes with one coin flip is now four possible outcomes with two coin flips. 101 00:07:40,790 --> 00:07:45,650 And if we express that probability in a table, here's what the table looks like. 102 00:07:45,650 --> 00:07:52,430 So again, we still have the same discrete random variable here where x is the number of times that 103 00:07:52,430 --> 00:07:53,600 we flip heads. 104 00:07:53,630 --> 00:07:59,600 Now that we're flipping the coin, two times we can flip heads zero times one time or two times. 105 00:07:59,600 --> 00:08:05,090 And the probability that each of these outcomes occurs is given here by the probability of X. 106 00:08:05,090 --> 00:08:11,540 We see that flipping heads zero times only occurs in this scenario here where we get tails tails. 107 00:08:11,540 --> 00:08:17,780 So there's a one in four chance that we flip heads zero times The probability that we flip heads two 108 00:08:17,780 --> 00:08:22,040 times occurs one out of four total possible outcomes. 109 00:08:22,040 --> 00:08:26,210 And that's here in this first scenario where we flip heads and then heads again. 110 00:08:26,210 --> 00:08:32,000 And there are two different ways that we could flip heads one time out of four possible outcomes. 111 00:08:32,000 --> 00:08:35,059 That's when we flip tails, then heads or heads than tails. 112 00:08:35,059 --> 00:08:42,140 So the probability of flipping heads one time is two matching outcomes out of four total equally likely 113 00:08:42,140 --> 00:08:42,770 outcomes. 114 00:08:42,770 --> 00:08:45,710 So we have here our probability distribution. 115 00:08:45,710 --> 00:08:49,550 And if we graph that, here's what that looks like. 116 00:08:49,550 --> 00:08:54,110 Now in our graph, we can flip heads zero one or two times. 117 00:08:54,110 --> 00:08:57,980 We can see that the probability of flipping heads zero times is one fourth. 118 00:08:57,980 --> 00:08:59,450 So we've sketched that in. 119 00:08:59,450 --> 00:09:02,720 The probability of flipping heads two times is also one fourth. 120 00:09:02,720 --> 00:09:09,800 So we sketch that in and the probability of flipping heads one time is two out of four or one half. 121 00:09:09,800 --> 00:09:15,350 So we sketch that in here and now this is the probability distribution for the discrete random variable 122 00:09:15,350 --> 00:09:20,420 x, where x is the number of heads that we flip when we flip a coin two times. 123 00:09:20,420 --> 00:09:22,580 So we're starting to get an idea here. 124 00:09:22,580 --> 00:09:28,040 Let's look quickly at what happens when we expand this to three coin flips and then four coin flips. 125 00:09:28,280 --> 00:09:32,630 So if we expand to three coin flips, here's what our table looks like now. 126 00:09:32,630 --> 00:09:34,520 We kept everything we had before. 127 00:09:34,520 --> 00:09:41,030 We said that this was our first coin flip in white, in blue was our second coin flip, and now in red 128 00:09:41,030 --> 00:09:43,970 we've attached an extra flip to those first two. 129 00:09:43,970 --> 00:09:47,960 And we've said that on this third flip, we flip heads every single time. 130 00:09:47,960 --> 00:09:53,840 But of course, we also have to account for the same set of the first two flips, but where on the third 131 00:09:53,840 --> 00:09:56,630 flip we flip tails every time. 132 00:09:56,630 --> 00:09:59,270 So now with three coin flips our. 133 00:09:59,540 --> 00:10:06,110 Were possible outcomes expands to eight possible outcomes, which means that our probability distribution 134 00:10:06,110 --> 00:10:12,410 table, if we're flipping the coin three times, we can flip heads 012 or three times. 135 00:10:12,410 --> 00:10:16,010 There are four different possible outcomes here. 136 00:10:16,010 --> 00:10:20,660 And if we just count up here, the number of times that each of those can occur. 137 00:10:20,690 --> 00:10:26,870 For instance, here, the only time that we flip heads zero times, it's when we get tails, tails, 138 00:10:26,870 --> 00:10:27,320 tails. 139 00:10:27,320 --> 00:10:32,780 So out of these eight possible outcomes, only one of them is flipping heads zero times. 140 00:10:32,780 --> 00:10:35,930 And so the probability there is one eighth same thing here. 141 00:10:35,930 --> 00:10:41,210 The probability of flipping heads three times is when we get heads, heads, heads, no tails. 142 00:10:41,210 --> 00:10:42,290 We see that here. 143 00:10:42,290 --> 00:10:45,440 That's one out of all eight of these outcomes. 144 00:10:45,440 --> 00:10:47,630 So the probability there is one eight. 145 00:10:47,630 --> 00:10:55,280 So we build out this table and now if we sketch this distribution in our graph, we see that it looks 146 00:10:55,280 --> 00:10:56,090 like this. 147 00:10:56,090 --> 00:11:02,210 What we're hopefully noticing at this point is that the distribution is always higher in the middle 148 00:11:02,210 --> 00:11:04,070 and lower around the edges. 149 00:11:04,070 --> 00:11:09,380 For all three of these scenarios where we were flipping a coin one time, two times, three times, 150 00:11:09,380 --> 00:11:12,740 the distribution has roughly the same shape every time. 151 00:11:12,860 --> 00:11:18,890 And if we flip a coin four times, so we add to this, we had our first flip, second flip, third flip, 152 00:11:18,890 --> 00:11:20,780 Now we add a fourth flip. 153 00:11:20,780 --> 00:11:26,960 So for all these scenarios here, we say that the fourth flip comes up as heads, and then we duplicate 154 00:11:26,960 --> 00:11:32,900 this whole set and say that the fourth flip this time comes up as tails every time. 155 00:11:32,900 --> 00:11:40,010 So our eight possible outcomes has now doubled to 16 possible outcomes, and our little probability 156 00:11:40,010 --> 00:11:42,500 distribution table looks like this. 157 00:11:42,500 --> 00:11:48,290 When we flip a coin four times, we can get zero one, two, three or four heads if we just count out 158 00:11:48,290 --> 00:11:54,500 of all 16 outcomes here, the number of times that we get zero heads one, two, three or four heads 159 00:11:54,500 --> 00:12:01,280 out of the 16, we can fill out here the probability of this specific number of heads. 160 00:12:01,280 --> 00:12:06,800 And if we sketch this distribution into our graph, it looks like this. 161 00:12:06,800 --> 00:12:13,490 And now we're really starting to see this sort of triangular shape of the probability distribution of 162 00:12:13,490 --> 00:12:15,290 this discrete random variable. 163 00:12:15,290 --> 00:12:20,750 And of course, if we kept going, this distribution would get more and more and more fine, more and 164 00:12:20,750 --> 00:12:25,670 more accurate, and we would see that the general shape of the distribution would look something like 165 00:12:25,670 --> 00:12:31,970 this, and it would approximate this curve more and more accurately as we add additional coin flips 166 00:12:31,970 --> 00:12:33,830 to our distribution. 167 00:12:33,830 --> 00:12:40,250 Now, all of that to say with this idea of a discrete random variable and its probability distribution, 168 00:12:40,250 --> 00:12:46,340 we want to recognize that discrete probability is just like any other probability in the sense that 169 00:12:46,340 --> 00:12:52,820 the probability of the discrete random variable is always going to sum to one or 100%. 170 00:12:52,820 --> 00:12:57,800 Remember earlier we said that probability was always defined between zero and one. 171 00:12:57,800 --> 00:13:03,320 And we can see here that all of these probabilities, all of these right hand columns in each of our 172 00:13:03,320 --> 00:13:13,190 four tables here for one coin flip, two coin flips, three coin flips and four coin flips, all of 173 00:13:13,190 --> 00:13:16,040 these probabilities are between zero and one. 174 00:13:16,040 --> 00:13:23,360 We have our smallest probability here at 116, all the way up to our highest probability of one half. 175 00:13:23,360 --> 00:13:30,770 All those values are between zero and one, and each of these right hand columns sums to one one half 176 00:13:30,770 --> 00:13:35,030 plus one half is one, which we could also call 100%. 177 00:13:35,030 --> 00:13:37,820 We could call each of these one half values 50%. 178 00:13:37,820 --> 00:13:40,940 So 50% plus 50% is 100%. 179 00:13:41,120 --> 00:13:48,170 Same thing here, one fourth plus two fourths plus one fourth is one or 100%. 180 00:13:48,170 --> 00:13:51,830 If we add up 25%, 50% and 25%. 181 00:13:51,830 --> 00:13:53,990 And that's going to be true for all of these. 182 00:13:53,990 --> 00:14:01,310 You'll see here that this adds up to eight over eight or one and that this adds up to 16 over 16 or 183 00:14:01,310 --> 00:14:02,000 one. 184 00:14:02,000 --> 00:14:06,320 So that concept still holds with discrete random variables. 185 00:14:06,320 --> 00:14:12,680 What we want to be able to do now is talk about how to calculate the expected value of a discrete random 186 00:14:12,680 --> 00:14:13,340 variable. 187 00:14:13,340 --> 00:14:19,250 And we also want to be able to look at variance and standard deviation for a variable X like we've been 188 00:14:19,250 --> 00:14:20,150 talking about. 189 00:14:20,150 --> 00:14:26,900 So expected value or that long term average is like what we talked about before when we looked at experimental 190 00:14:26,900 --> 00:14:32,240 versus expected probability and to calculate it here for a discrete random variable. 191 00:14:32,240 --> 00:14:35,750 So let's take this scenario where we have just one coin flip. 192 00:14:35,750 --> 00:14:41,510 When we have a discrete random variable, we have to take what's essentially a weighted average. 193 00:14:41,510 --> 00:14:45,980 So we look at here these different countable outcomes. 194 00:14:45,980 --> 00:14:51,770 X, we say that when we flip a coin one time, we can either flip heads zero times or one time, and 195 00:14:51,770 --> 00:14:54,920 the probability that each of those things occurs is one half. 196 00:14:54,920 --> 00:14:59,150 So we take the possible outcomes here zero and one and we. 197 00:14:59,250 --> 00:15:02,430 Wait each of them by their probability, one half. 198 00:15:02,430 --> 00:15:07,600 And when we do the math here, we get zero plus one half or one half. 199 00:15:07,620 --> 00:15:14,520 And what this tells us then, is that when we flip a coin one time, the expected value for the number 200 00:15:14,520 --> 00:15:17,420 of times that we flip heads is one half. 201 00:15:17,430 --> 00:15:25,380 So it's almost saying that we can expect half heads on our one coin flip, which of course doesn't make 202 00:15:25,380 --> 00:15:32,160 sense when we flip a coin one time, we know that we can only flip heads zero times or one time we can 203 00:15:32,160 --> 00:15:34,140 never flip it half of a time. 204 00:15:34,140 --> 00:15:37,140 But this is what the expected value is saying. 205 00:15:37,140 --> 00:15:42,990 And of course, in a way that makes sense because if we're trying to represent the number of times we 206 00:15:42,990 --> 00:15:49,950 should expect heads on one coin flip, even though we know the only outcomes are zero or one, it would 207 00:15:49,950 --> 00:15:56,340 be wrong to say that our expectation on one coin flip is always zero heads, or that our expectation 208 00:15:56,340 --> 00:15:58,650 on one coin flip is always one heads. 209 00:15:58,650 --> 00:16:01,200 We can't really use either one of these values. 210 00:16:01,200 --> 00:16:06,510 We have to go in the middle because as we already know, we've looked at it already when we talked about 211 00:16:06,510 --> 00:16:07,350 probability. 212 00:16:07,350 --> 00:16:09,930 And we also should just know this intuitively. 213 00:16:09,930 --> 00:16:12,930 We should expect to flip heads about 50% of the time. 214 00:16:12,930 --> 00:16:19,110 And so if we're flipping a coin only one time in this theoretical world here, we should expect half 215 00:16:19,110 --> 00:16:21,780 of a heads on that one flip. 216 00:16:21,780 --> 00:16:26,430 Let's look at what happens, though, when we flip a coin two times, three times, four times to our 217 00:16:26,430 --> 00:16:27,450 expected value. 218 00:16:27,660 --> 00:16:30,960 So when we flip a coin, twice are expected. 219 00:16:30,960 --> 00:16:38,220 Value is this We have all of our different outcomes zero one or two heads, and we wait them with their 220 00:16:38,220 --> 00:16:39,990 associated probabilities. 221 00:16:39,990 --> 00:16:47,340 And so when we do the math here, the result is zero plus one half plus one half or one. 222 00:16:47,580 --> 00:16:53,130 And this then should make a lot more sense to us because we're saying if we flip a coin two times, 223 00:16:53,130 --> 00:16:55,770 we expect to flip heads one time. 224 00:16:55,770 --> 00:16:57,570 That is our long term average. 225 00:16:57,570 --> 00:17:01,620 That is our expected value for this discrete random variable. 226 00:17:01,620 --> 00:17:04,020 And of course, that makes sense to us. 227 00:17:04,020 --> 00:17:11,190 What happens, though, when we flip a coin three times, this time we can flip heads zero times once, 228 00:17:11,190 --> 00:17:12,630 twice or three times. 229 00:17:12,630 --> 00:17:20,250 We weight these values with their associated probabilities and we get zero plus three over eight plus 230 00:17:20,250 --> 00:17:26,069 six over eight plus three over eight, and we get 12 over eight or three halves. 231 00:17:27,359 --> 00:17:27,990 Three halves. 232 00:17:27,990 --> 00:17:30,200 Or we could think about this as one and a half. 233 00:17:30,210 --> 00:17:35,730 So what this is saying is that if we flip a coin three times, we should expect one and a half heads, 234 00:17:35,730 --> 00:17:40,830 which of course in the context of the discrete random variable where we know it can only take on the 235 00:17:40,830 --> 00:17:46,710 values zero one, two and three one and one half heads doesn't make sense. 236 00:17:46,710 --> 00:17:53,610 But as a long term average, of course it does make sense because we know that flipping a coin, we 237 00:17:53,610 --> 00:17:58,800 should always get 50% heads, 50% tails as long as we're taking a big enough sample, as long as we're 238 00:17:58,800 --> 00:18:02,790 flipping the coin enough times to get a good long term average. 239 00:18:02,790 --> 00:18:07,590 And so if we're flipping the coin three times, of course, we should expect one and one half heads. 240 00:18:07,590 --> 00:18:13,410 And then the last one, hopefully you can already guess by now, when we flip the coin four times, 241 00:18:13,410 --> 00:18:20,340 we see that we have all the possible outcomes zero one, two, three or four head flips and we weight 242 00:18:20,340 --> 00:18:27,930 them with their associated probabilities and we get zero plus four over 16, plus 12 over 16 plus 12 243 00:18:27,930 --> 00:18:35,130 or 16 plus four over 16, and we end up with 32 over 16 or just two. 244 00:18:35,130 --> 00:18:40,770 And that makes sense because when we flip a coin four times, we should expect long term on average 245 00:18:40,800 --> 00:18:43,020 heads to come up two times. 246 00:18:43,020 --> 00:18:46,140 So in a way, all of this should make sense. 247 00:18:46,140 --> 00:18:53,430 But the real unlock here shows up in these odd values where we're flipping a coin one time or we're 248 00:18:53,430 --> 00:18:59,280 flipping a coin three times the even values or flipping a coin two times, four times, six times, 249 00:18:59,280 --> 00:19:01,350 etc. Those make sense to us. 250 00:19:01,530 --> 00:19:08,220 We get these nice whole round numbers where the expected value is one, two, etc. But what these odd 251 00:19:08,220 --> 00:19:15,450 values show us is that we can get an expected value, which is in fact our mean. 252 00:19:15,450 --> 00:19:23,460 We call this the mean of the discrete random variable x, this is mu sub x and it represents the mean 253 00:19:23,460 --> 00:19:26,580 and in the case of a discrete random variable, it's the expected value. 254 00:19:26,580 --> 00:19:34,110 But these odd values show us that the mean of a discrete random variable can turn out to be a value 255 00:19:34,110 --> 00:19:38,820 that is in between the discrete values that the variable can actually take on. 256 00:19:38,820 --> 00:19:45,210 So even though here this variable can only take on values of zero or one, we find a long term average 257 00:19:45,210 --> 00:19:51,810 of one half, even though here with three coin flips, the discrete random variable can only take values 258 00:19:51,810 --> 00:19:53,760 of zero one, two or three. 259 00:19:54,000 --> 00:19:55,620 This expected value formula. 260 00:19:55,620 --> 00:20:02,790 This idea shows us that we can still calculate a mean even for a discrete variable where that mean ends 261 00:20:02,790 --> 00:20:07,710 up being a value in between the values that the variable can actually assume. 262 00:20:07,710 --> 00:20:13,860 So even though here we can only find values of zero one, two or three, the mean is three halves. 263 00:20:13,860 --> 00:20:15,420 The mean is one and one half. 264 00:20:15,420 --> 00:20:18,060 Not any of these values at all. 265 00:20:18,450 --> 00:20:25,140 So we understand now the idea of the probability that a discrete random variable takes on a particular 266 00:20:25,140 --> 00:20:25,770 value. 267 00:20:25,800 --> 00:20:30,960 We've seen what that looks like in terms of the distribution, the probability distribution of that 268 00:20:30,960 --> 00:20:32,280 discrete random variable. 269 00:20:32,280 --> 00:20:38,670 And we understand now how to find the expected value or the mean of that discrete random variable. 270 00:20:38,670 --> 00:20:45,420 And that the mean may be a value that is between some of the values that the discrete random variable 271 00:20:45,420 --> 00:20:47,070 can actually assume. 272 00:20:47,250 --> 00:20:53,760 So now that we have this mean, this expected value, we want to just talk about the variance and standard 273 00:20:53,760 --> 00:20:56,850 deviation of a discrete random variable. 274 00:20:56,850 --> 00:21:01,140 And this is going to be our variance formula for a discrete random variable. 275 00:21:01,170 --> 00:21:06,510 These x sub values are just the individual values of the discrete random variable. 276 00:21:06,510 --> 00:21:10,320 In this case, with four coin flips, that's zero one, two, three and four. 277 00:21:10,320 --> 00:21:14,370 And then you'll see here the probability of each of those values. 278 00:21:14,370 --> 00:21:20,250 So this little probability distribution table gives us all the information we need to substitute for 279 00:21:20,250 --> 00:21:24,600 x abi and the probability of x sub i and then mu. 280 00:21:24,630 --> 00:21:28,110 Here is the expected value that we looked at earlier. 281 00:21:28,110 --> 00:21:28,980 It's the mean. 282 00:21:28,980 --> 00:21:34,950 And so you can see what we're doing is just finding the difference between the value of the discrete 283 00:21:34,950 --> 00:21:36,930 random variable and the mean. 284 00:21:36,930 --> 00:21:38,970 And then we're squaring that distance. 285 00:21:38,970 --> 00:21:45,780 Squaring the distance is going to guarantee that we get a positive outcome here for this x sub I minus 286 00:21:45,780 --> 00:21:47,070 mu quantity squared. 287 00:21:47,070 --> 00:21:52,080 So this value is always going to be positive and then we're going to multiply that by the probability 288 00:21:52,080 --> 00:21:59,490 over here P of X and we're going to do that for each one of the possible values of X, and then we're 289 00:21:59,490 --> 00:22:01,890 just going to add them all together. 290 00:22:01,890 --> 00:22:05,730 That will give us the variance of this particular discrete random variable. 291 00:22:05,730 --> 00:22:11,340 So let's do that for the discrete random variable x here where x models the number of times we flip 292 00:22:11,340 --> 00:22:13,890 heads when we flip a coin four times. 293 00:22:13,890 --> 00:22:20,730 So if we substitute, our variance here will be our first value of x, which is zero. 294 00:22:20,730 --> 00:22:23,790 So we'll say zero minus the mean. 295 00:22:23,790 --> 00:22:26,790 We calculated the mean down here and we said that that was. 296 00:22:26,900 --> 00:22:27,310 Two. 297 00:22:27,310 --> 00:22:33,760 So we get zero minus two quantity squared times the probability of getting zero heads when we flip the 298 00:22:33,760 --> 00:22:36,520 coin four times, which we know is one out of 16. 299 00:22:36,520 --> 00:22:44,800 So we multiply this by one over 16 and then we keep going until we get to the bottom of our probability 300 00:22:44,800 --> 00:22:47,770 distribution chart here with x equals four. 301 00:22:48,100 --> 00:22:56,500 So here our next row where X is one, we flip heads one time we get one minus two quantity squared times 302 00:22:56,500 --> 00:22:58,060 four over 16. 303 00:22:59,610 --> 00:23:06,960 And then we add to that when X is two here, two minus two quantity squared times the probability, 304 00:23:06,960 --> 00:23:08,790 they're six over 16. 305 00:23:10,120 --> 00:23:18,220 And then we add to that three minus two quantity squared multiplied by four over 16. 306 00:23:18,220 --> 00:23:26,840 And then the last row here plus four minus two quantity squared times one over 16. 307 00:23:26,860 --> 00:23:30,550 And of course, we're doing this by hand just so we can see the math. 308 00:23:30,550 --> 00:23:37,930 But all modern scientific calculators and computer programs will calculate a variance for you based 309 00:23:37,930 --> 00:23:40,120 on this kind of probability distribution. 310 00:23:40,120 --> 00:23:45,820 So in reality, we're not ever doing this math by hand, but we just want to see what's actually going 311 00:23:45,820 --> 00:23:47,800 on behind the scenes so we understand what's happening. 312 00:23:47,800 --> 00:23:51,040 So this is the manual calculation. 313 00:23:51,040 --> 00:23:53,530 And when we actually do this here, we get negative two. 314 00:23:53,530 --> 00:24:02,440 Quantity squared is four, four times one over 16 is four over 16, one minus two is a negative one, 315 00:24:02,440 --> 00:24:04,300 quantity squared is positive one. 316 00:24:04,300 --> 00:24:09,190 So plus four over 16 two minus two is zero. 317 00:24:09,190 --> 00:24:10,990 So that whole term is going to drop away. 318 00:24:11,020 --> 00:24:19,300 Three minus two is one squared is one times four over 16 is four over 16 and then four minus two is 319 00:24:19,300 --> 00:24:19,750 two. 320 00:24:19,750 --> 00:24:24,820 Quantity squared is four times one over 16 is four over 16. 321 00:24:25,000 --> 00:24:28,870 We end up with four plus four plus four plus four is 16. 322 00:24:28,990 --> 00:24:33,430 So we get 16 over 16 or one. 323 00:24:33,550 --> 00:24:40,120 And so our variance for this particular discrete random variable, the number of heads we flip when 324 00:24:40,120 --> 00:24:42,790 we flip a coin four times is one. 325 00:24:42,790 --> 00:24:49,120 And as we already know, the standard deviation we can find by taking the square root of variance. 326 00:24:49,120 --> 00:24:56,080 So the standard deviation of this discrete random variable x will be the square root of variance or 327 00:24:56,080 --> 00:24:58,570 the square root of one or just one. 328 00:24:58,600 --> 00:25:02,770 Of course, these numbers worked out really cleanly for this particular variable. 329 00:25:02,770 --> 00:25:04,390 That won't always be the case. 330 00:25:04,390 --> 00:25:11,380 But now, even though we started with a discrete random variable, we can say that this particular discrete 331 00:25:11,380 --> 00:25:14,560 random variable x has an expected value. 332 00:25:15,440 --> 00:25:27,170 Or a mean of two, that it has a variance of one in a standard deviation of one. 333 00:25:27,170 --> 00:25:33,800 And now we can use all this information to answer probability questions about the discrete random variable, 334 00:25:33,800 --> 00:25:42,290 like what is the probability that we get more than 3.25 heads when we flip a coin four times, or the 335 00:25:42,290 --> 00:25:48,440 probability that we flip some number of heads between 1.5 heads and 2.7 heads. 336 00:25:48,440 --> 00:25:51,650 And we're not bound by these discrete values. 337 00:25:51,650 --> 00:25:57,440 We've sort of, with these calculations, transformed what would otherwise be a completely countable, 338 00:25:57,440 --> 00:26:02,150 discrete variable into something where we can examine the values in between.