in this video we want to start our conversation by talking about distributions distributions in probability means we're going to be examining the probabilities of all of our outcomes of a certain kind let's go back to those two six-sided dice here's our table that represents all the possible ways the dice can show but if we want to talk about only the sums of these dice right one and one is 2 3 and 2 is 5 6 and 6 is 12 and so on but I want to talk about all of the probabilities for all of the possible sums we're going to get something like this where along the bottom in the red numbers those are the possible sums we can get 2 3 4 5 all the way to 12 the numbers at the top are going to represent how many times each outcome can occur if we look at the table everything highlighted in blue adds up to six and there are five of those objects the white numbers at the bottom represent the probability of each of these sums happening as a percentage this right here is our distribution the probability that X will come up the sum of the two dice will come up as a 2 3 4 5 Etc is given by this table right here I look at the sum that I want and I find the probability inside the distribution if we want to build a distribution it's kind of just like doing extra probability in the problem below we're going to create a distribution for this random variable the experiment being having three kids and recording the number of girls since I'm recording only the number of girls while I need the sample space all I care about is how many girls are in each configuration so the distribution starts with our outcome the number of girls I can have zero girls one two or three to calculate the probability we'll go ahead and build our sample space and notice we have a situation where there are no girls a situation where there's only one we've got three of those a situation where there are two girls we have three of those as well and a situation where we have all three girls well there are eight things in the sample space so for each possible outcome 0 1 2 or three I find its probability the probability of no girls is 1 out of eight then three out of of eight 3 out of 8 and 1 out of 8 we want to talk about distributions so that we can talk about expected value that is if I do the experiment an infinite number of times what is the average value of the experiment so start with this game described below we have a bag of marbles we've got two gold 10 silver 30 black and the following game is going to be played you pick a marble out of the bag and if it's gold marble that's two of them you win $5 if it's a silver one you'll get $2 and if it's a black one you lose $2 so what is the expected value if you play the game now make sure that we're understanding here is expected value here is saying an infinite number of times what does that even mean so let's break this down really slowly first what are our outcomes we can win $5 $2 or we can lose $2 and then we need the probability so that we can get our distribution well if we count up all the marbles we have 42 of them total that's the number of things in our sample space which means that the probability of getting $5 out of the bag that's going to be two out of 42 two gold marbles the probability of getting a silver one out of the bag that that'll be 10 of 42 and for losing $2 that'll be 30 over 42 now we always want our fractions reduced but how do we find an average you can only play the game once right I draw the marble I do what it says so the average is going to work like this and this is where the infinite part comes into play if I play game one and I get $5 that's pretty sweet I won the first time around now let's play game two so immediately after winning the $5 I'm going to put the marble back I'm going to shake the bag up and I'm going to pull another marble but this time it's silver so I only get $2 well I just played two games back to back what was the Val what was the prize money per game that's the average so 5 + 2 divided by two $5 $2 divided by two games that's 7 over2 $3.50 now I put the marble back and we play again but I get a black marble feed it into the average now I've got 5 + 2 minus 2 over 3 that's 5/3 or a167 now we put the black marble back and we shake it up and we play it again I lose $2 again that brings my average down to 75 cents now there's something very important to notice here I can keep doing this forever I mean obviously not literally forever but every time I play the game the outcome is going to change the prize money per game that I'm playing something important to address here you are never going to end up with winning exactly $33.50 by playing this game you're never going to end up with exactly a167 or any of the other intermediate calculated averages in fact even the expected value when we Define it you cannot end up with that average and you can't end up with that amount of prize money the prize money is only going to be a combination of how many times you got $5 how many times you got $2 and how many times you lost $2 those are all whole dollar amounts you're never going to get change what the expected value wants to tell us is the average value of playing the game and this is just over the long term and this is going to help us make decisions about the thing that we're looking at so how do we calculate the expected value we're going to use the probability distribution because that's how things should shake out over the long term and we're going to combine it with the values the prize money from each of those outcomes so we'll Define this as e with the X in parenthesis e of x equal to the first probability times its value plus the second probability times its value and so on until we've gone through the entire distribution of the experiment a quick calculation shows that we're going to get a value of negative 0.71 so the game has an expected value of losing 71 cents if you keep playing over and over and over again in the long term now you might ask yourself why on Earth would you want to play a game an infinite number of times where you're losing something well don't think of a game as something you do for fun but think of a game the same way we've been thinking of the word experiment what's something that you want to know about what's something that you want to be able to do over and over and over the one place you're playing an expected value game every day is with your Insurance homeowners insurance insur renters insurance car insurance it doesn't matter let's take a small example so the numbers are easier to work with if we have a bicycle insured against theft or any other loss with a $600 policy for an annual premium of $55 and the probability that that bicycle is going to be stolen or otherwise lost is 0.08 that is an 8% um certainty that it will be stolen what's the expected value of the policy well what are our outcomes loss no loss the bike is stolen or it's not it's total or It Isn't So what are the values well in the case of loss I'll get $600 however I already paid 55 in as my premium so I'm only going to actually get $545 the value of no loss well that's me paying my premium and nothing happening then we look at the probabilities well we were told the probability of theft was 8% or 0.08 quick note you do have to use decimal forms here you cannot use percentages fractions are okay as well so if the probability of loss is 0.08 what's the probability of no loss that's going to be 0.92 remember the rule of one if the event is loss then the complement is no loss and we just subtract that probability from one to get the probability of no loss so the expected value will be the 8% * 545 plus the 92% * 55 giving us a total of minus 7 that means that the expected value is going to be a loss of $7 and if you think about it that's kind of what's happening when you have your car insurance how much you paying per year per six months per month right a standard car insurance policy for the minimums around this area of Texas you're spending out maybe1 or $2,000 a year how often are you claiming the 10 25 or $50,000 coverages on your policy almost never so why have insurance in the first place because I can afford the negative $2,000 every year I cannot afford to lose $50,000 once so even if the expected value of a game is negative it still might be some something that you're interested in playing so given a probability distribution a bunch of probabilities P1 P2 P3 with outcomes X1 X2 X3 the expected value of the distribution is the sum of each outcome multiplied by its probability so capital E of X is going to be X1 * P1 right the outcome times its probability plus X2 * P2 plus X3 * P3 and so on and so on remember your order of operations make sure you do your multiplications before your additions that's a good place to stop