Lecture 07

[Music] in this discussion you'll be introduced to the concept of probability distributions specifically discre distributions and their probabilities first we'll talk about random variables and probity distributions if you recall an experiment is an any process that leads to one of seven possible outcomes so the random variable is just a mathematical way of describing these outcomes and when we plot the probability of all the random variables on a graph then this graph is what we call probability distribution so the random variable and probability distribution are two related Concepts and we'll learn about both of them in much more detail the next concept that we'll see is that of expected value and variance this is very similar to what we calculated in the form of mean and variance for a population or a sample the only difference being that you'll learn how to calculate the same for a probability distribution expected value is generally used to find an anticipated value for an investment at some point in the future and is popularly used in Industries like insurance and card games we'll see some interesting applications around the expected value apart from that you'll also learn about the important discrete probability distributions that might come across in your day-to-day applications more specifically the binomial distribution binomial distribution could be used to mimic experiments like public votes in an election survey or analyzing new products response via a limited launch random variables part one I'm quite sure that you have heard this well-known saying about casinos that The House Always Wins what it means is that the casino gambling machines and games are designed in such a way that in the long run casinos always make money one or two people may end up winning large sums of money but the machines are designed such that the rest of the people collectively lose more so how do the casinos ensure that the house always wins it is quite simple they use probabilities in fact casinos use a very basic concept of probability to ensure a win for themselves do we wonder how to understand this let me tell you about a simple gambling game we can play we can take a bag and fill it up with three balls two red and one blue each parent had to take out a ball not its color and then put it back in this process was followed until each parent has taken out a ball from the bag four times a parsent who got a red ball all four times would receive 150 rupees from the house attractive offer isn't it however the parsan who got any other result would have to put or pay 10 Rupees so would you play or would you like to play this game is this the kind of game in which we the house should always win in the long run or are we going to incur some losses in the long run in order to understand all this we'll approach the problem in three steps first we'll see what all kinds of combinations can come up then we'll see how likely each of these combinations that is what their probability is then we'll see how we can use these probabilities to estimate the profit and loss of a player he would get on playing the game once let's go through the various outcomes possible after apartent has tried the BET as you would guess the one outcome our players would want the most is that all four balls end up being red however that is not the only possible outcome they could for example get three red balls in a row only to find that the last ball is sadly a blue one wouldn't that be sad so can you find out all the possible outcomes so let's first list down all the possible outcomes one by one we could get four blue balls there's only one outcome in which this happens we could get three blue balls and one red ball this could happen in four ways rle B BRB double BRB and triple BR we could also get two blue balls and two red balls this could happen in six ways also we could get one blue ball and three red balls in four ways and there's only one way in which we could get four red balls so in total There are 16 possible outcomes random variables part two let us go back to some of the original questions that we were concerned with starting with How likely is the house to win or what is the profit or loss you should expect while running such a business however currently we cannot answer these questions and in order to answer them we need to analyze the outcomes in terms of probabilities for example if the probability of getting four red balls in a row comes out to be extremely low we may say that the house is likely to win on the other hand if the probability of getting four red balls is comparable to the other possible outcomes it may be profitable for the players let us think about how we can do that so there are various ways in which we can convert these outcomes to numbers or in other words quantify them let's assign the value of the number we get after quantifying to a variable X for example let's quantify the outcomes by using the number of red balls so instead of saying that the outcome is blue red red red we can say that we got three red balls and x equal to 3 similarly for the outcome blue red blue red we can say x equal to 2 as you can see this number used for quantifying outcomes is really important in statistical language it is known as random variable and is denoted by uppercase X it is basically any variable that converts an outcome to a number another important question is are there more ways in which random variable X could be defined we can Define X as the number of blue balls that we drew from the back or the number of red balls minus the number of blue balls that we have drawn from the bag so the question that now arises is which is the correct random variable out of all these options that you have which one should you choose well the answer actually depends on the information you are interested in in our example we are interested in the number of red balls recall that if the player draws four red balls he or she wins 150 rupees however if the player draws even 3 two 1 or zero red balls he or she loses 10 Rupees so let's define X this way x equal to number of red balls now if you want to find whether we will actually make money or lose it we need to know how likely the different values of X are what is the probability of each value we'll see that in the subsequent discussions probability distributions let us see the different values of the random variable X as we defined before it will take for all the possible outcomes of our game remember that X was defined as the number of red balls that we have drawn from the back so if the possible outcome that we look at is the one with four blue balls obviously X takes the value zero for all four outcomes where we only Drew one red ball out of the four x will take value of one for the six possible outcomes where there are two red balls and two blue balls in the various combinations X will take a value of two for these four outcomes where we have three red balls and one blue ball X will take the value three and for the last outcome where we have four red balls X takes the value of four so basically by doing this we have brought our 16 outcomes into five groups where the random variable takes values from 0 to 4 now if you want to find out whether we'll make or lose money in the long run need to find out the likelihood of each of these values occurring but how to do that let us say that we conducted experiment with certain 100 individuals who played this gamble based on each person's response we put a small note on a chart for example person a got three red balls so we put a note with his name on the chart over value three similarly person B got two red balls so we put a note with his name over the value two this way as more and more people play we will fill up the chart accordingly until we have responses of all the 100 people on this chart so in this chart over here on the x-axis we have random variable X from value 0 to 4 on the y axis we have the frequency the number of times the random variable appeared in our experiment so this chart is called a histogram so the heights of these bars in this blot correspond to the number of times the frequency of that random variable appeared in our experiment for example two out of the 100 people drew no red balls 10 out of the 100 people drew one red ball and so on so recall that the probability of any outcome is the number of favorable outcomes divide by the total number of outcomes so let's find the probability that our random variable X takes the value two the number of favorable outcomes in this case will be the number of people who drew two red balls and that number is 32 the total number of possible outcomes is 100 as that is the total number of people who participated in that game so the probability of drawing two red balls is 32 by 100 which is equal to 0.32 or 32% so we can extend this probability calculation for all the different values of X and put all of this information in a single table as shown here after calculating the probabilities for X = to 0 1 2 3 and 4 we get the probability that X is equal to 0.02 the probability that the random variable X takes a value of 1 is 0.07 the probability that X takes value of 2 is 0.32 probability that X takes the value 3 is 0.43 and for xal to 4 the probability is 0.16 now this table here is called the probability distribution as it tells us the probability of all the possible values of X so now we will take these probability values and just draw them in a chart to a our understanding of the probability values on the x-axis here we have five random variable values and on the y axis we have our probability values the chart here where we plot our probability value Val versus the random variable values is called the probability distribution chart so as you can see the probability distribution when displayed in the form of a bar chart look exactly like a original histogram just with a different scale that is since probabilities can lie between 0 and one only the values on the y- axis are not restricted in the range of 0 and one each value here represents the probability of getting a certain number of red balls so this is what our probability distribution looks like in tabular form and in bar chart form as you can see clearly from the this example a bar chart gives us a much better visual idea of what the probability will be hence this form is more commonly used now we can use this distribution which gives us How likely each value of x is and find the answer to our original question that is would be in Long Run make money or lose money so if we recall from our office game setup we fixed our random variable X as the number of red balls we got in the experiment we then made a probability distribution for X which will help us to see how likely each of these values of X is to happen now as we said earlier we want to use probability and find whether in the long term the game or a bet will be profitable to us the house or not question is how will we do that let us see suppose thousand people are playing this game how much do you think an average person would win or lose in this game can you try and take a guess so for these thousand people do you think you'll be able to tell me how many people are expected to draw one red ball how many people are expected to draw two red balls three red balls and four red balls as you remember from before we know that the probability that X is equal to 1 is the value 0.07 so multiplying that by thousand will give us the expected number of players who will draw one red ball th000 * 0.07 gives us 70 so in other words about 70 people will draw one red ball out of those four balls similarly the probability that X is equal to 2 will be 0.32 so multiplying by 1,000 gives you 320 so again it means that 320 players on average would draw two red balls similarly for xal to 0,000 time 0.02 which gives us 20 people who will draw zero red balls 430 people will draw three red balls and 160 will draw four red balls in other words 160 people on average will win this game question is so how many red balls are we expecting to see on average in our game remember thousand people played this game and out of those thousand 70 people had drawn one red ball similarly 320 people drew two red balls zero balls were drawn for 20 people three balls by 430 people and four balls by 160 people so I can say that the total number of red balls we'll get after th000 attempts at the game will be 0 into 20 1 into 70 2 into 320 3 into 430 + 4 into 160 the sum of all these events gives us 2640 red balls so basically we saw 2640 times a red ball in our, experiments so if we take the division of these two we can say that we are seeing 2.64 red balls in one experiment on average in other words we can expect a person to draw 2.64 red balls in one game and so in net shell the average value that we would expect to get for a random variable is called its expected value let's continue our discussion of expected value let's now formally Define the expected value let's say that a random variable X can take values X1 X2 X3 and so on till xn the expected value of this random variable X would be given by the expression expected value equal to X1 * probability of X1 plus X2 * probability of X2 plus X3 * probability of x3 and so on till xn * pxn and remember probability of x equal to x i denotes the probability that the random variable X takes the value x i in our example the value that the random variable X can take 0 1 2 3 and 4 so the expected value of this random variable would be given by the expression expected value equal to 0 into probability of 0+ 1 into probability of getting XS 1 2 into probability of getting XS 2 + 3 into probability of getting X as 3 + 4 into probability of getting X as 4 we have already computed these probabilities before and plugging those values we get the expected value as 0 into 02 + 1 into 07 + 2 into 0.32 + 3 into 0.43 + 4 into 0.16 = to 2.64 so again the expected value for random variable X that is the number of red balls is 2.64 however I'm sure you are wondering how can you get 2.64 balls in one attempt the number of Falls would be two or three why are we saying that on average we expect to get 2.64 balls actually it is not unusual to have an expected value that may never actually come up we say that the expected value is 2.64 it doesn't mean that we expect to get red balls in an attempt that that many 2.64 balls what do this value actually means is that if you were to play a game infinite number of times then the average number of red balls you would get is 2.64 but does this value really even help us with our question we still can't tell whether we would make money or lose in the long run so what should we have done differently to answer our primary question if you want to find out whether the house would make or lose money in the long run then we would need a different random variable probably a better random variable here would be the amount one after playing the game let's continue our discussion of expected value so let's define our random variable this way x equal to money one after playing the game once this definition of X works well for us if we get a high expected value for it that would imply that on average players will win a lot of money in our game which is obviously not a good thing for us on the other hand a low expected value here would mean that on average the players don't p a lot of money so this new random variable X can take to values minus 10 and + 150 this because player wins $150 for getting four red balls and loses $10 for getting any other outcome we know that the probability when X takes the value 150 is the probability of getting four red balls is equal to 0.16 on the other hand the probability of losing the game when the random variable would take the value minus 10 is 0.84 which is the probability of getting any other possible combination of balls namely getting 0 one two or three red balls so we can go ahead and compute probabilities of getting 0 1 2 3 red balls we could get the probability of getting 0 1 2 3 red balls by adding these individual probabilities 0.02 + 0.07 + 0.32 plus 0.43 which is equal to 0.84 in other words the probability that a player will lose this game is 0.84 applying the same definition of expected value as we saw before we can compute the expected value for this event as 115 to 0. 6us 10 into 0.84 which is equal to $15.6 what this means is that on average a player would expect to win 15.6 by playing the game so it turns out that the game is a disaster for the house if you want to make money you need to ensure that the expected value for the money won by a player is negative this is to ensure that players lose money and the house wins in the long run so how can you do this you can decrease the price money increase the penalty or just decrease our players chances of winning any of these choices will decrease the players expected earnings from the game we can go ahead and fill in our rules and configuration of the game so you can now understand how casinos ensure that they remain profitable the casino machines are all designed in such a way that the expected value of the money earned by the player is always negative this means if we design the game such that the expected value of the money earned by a player is for example minus10 orus $20 we can be fairly confident that the house will be profitable let's look at one real life example of the expected value we know that the expected value is the basis for insurance companies and we can do a simple example based on this concept based on certain sources the company estimates that the probability that an accident will occur within the next year is 0.071 on the basis of this information What premium should the insurance company charge to break even on a $400,000 on a one-year Term Policy so let us try to calculate the expected value that the company will need to pay to settle a policy in case of an event of an accident the probability is 0.071 and the amount to be paid is $4 lakh however in case of no accident the probability becomes 1 minus 0.71 which is equal to 0.929 and in this case the amount to be paid is zero thus the expected value will be x * probability of X when there is no accident plus X time probability of x when an accident occurs in case of no accident x * probability of X becomes 0 * 0.929 which is 0o as shown in the table here and in case of an accident X time probability of X becomes 0.71 * $4 lakh which is approximately $284 these computations are provided in the table thus on average the company needs to pay about $284 to settle a policy thus the premium should be $284 plus whatever return the company needs to cover such administrative overhead and profits variance we can calculate the expected value if we are given a random variable and its corresponding probity distribution so this answers our question about the central tendency of the distribution however you also learned that apart from the central tendency we also look at the spread or variability of the data so let's go back to some of our previous example where we created a probability distribution of the random variable X where X is the number of red balls in the bag we want to find out the variance of X to calculate the variance we have a very simple formula for doing so basically it is variance of xal to xus Mu to^ 2 * probability of X summed over all the values of X or alternatively summation of x - mu^ 2 into probability of x equal to x i where mu or sometimes called as xar or expected value of mean of the random variable recall that for xal to 0 the probability was 0.02 for xal to 1 it was 0.0 7 for x = 2 it was 0.32 and for x = 3 it was 0.43 and for xal 4 it was 0.16 here mu denotes the expected value that we had earlier calculated so it will be the expected value of the number of red balls which we calculated to be 2.64 so to calculate the variance of X it will be 0 - 2.64 to ^ 2 into 0.02 + 1 - 2.64 ^ 2 into 0.07 + 2 - 2.64 ^ 2 into 0.32 + 0 - 2.64 ^ 2 * 0.43 + 4 - 2.64 ^ 2 into 0.16 which sums up to 0.81 04 thus the variance of this distribution is 0.814 and the standard deviation will be the root of 0.81 04 which would be 0.92 calculating the variance could be used to compare the spread of different probability distributions [Music] [Music] [Music]