Expected Values, Main Ideas!!!

today we're gonna talk about what to expect when you're expecting a value stat quest hello i'm josh starmer and welcome to statquest today we're going to talk about expected values and they're going to be clearly explained let's start by taking a trip to the magical place called statland once we get to statland our friend statsquatch says hey i bet the next person we meet has heard of the movie troll 2 and we say are you kidding me troll 2 is one of the worst movies ever made why would anyone know about it and then stat squash says i bet you one dollar that the next person we meet has heard of the movie troll 2. so we think to ourselves good thing we recently asked everyone in statland if they'd heard of the movie troll 2. here's the data there are two types of people in statland people that have heard of the 1990 movie troll 2 and people who have never heard of troll 2 in other words there isn't anyone who has both sort of heard of troll 2 and sort of hasn't heard of troll 2. each person in statland is in one group or the other 37 of the people in statland have heard of troll 2 and the remaining 176 people in statland have never heard of troll 2 that means there are 213 people living in statland so statland is not a huge place looking at the raw numbers is useful because they tell us that statland is pretty small and that whatever analysis we do only applies to a handful of people that said by looking at the numbers we can get a general sense of the trends in statland in this case we see that most of the people in statland have never heard of troll 2 but at a glance it's not super obvious how the number of people who have never heard of troll 2 relates to the total population of statland the good news is that we can make this relationship super obvious by calculating probabilities so let's calculate the probabilities of people in statland that have and haven't heard of troll 2 if we want to know the probability that a randomly selected person in statland has heard of troll 2 we simply divide the number of people that have heard of troll 2 37 by the total 213 and that gives us 0.17 that means the probability that a randomly selected person in statland has heard of troll 2 is 0.17 which is relatively low likewise the probability that a randomly selected person in statland has never heard of troll 2 is 0.83 which is relatively high now remember that bet our friend statsquatch wanted to make i bet you one dollar that the next person we meet has heard of the movie troll 2. the good news is that we can use these probabilities to decide if we should agree to the bet so let's move the probabilities to the top so we can focus on them now if the next person we meet has heard of troll 2 then we will lose the bet and that means we will lose one dollar so let's put negative one here to represent the outcome of losing one dollar if we meet someone who has heard of troll two in contrast if the next person has not heard of troll two then we will win the bet and that means we will win one dollar so let's put one here to represent the outcome of winning one dollar if the next person we meet has not heard of troll 2. so the left side of this table tells us that the probability we will lose one dollar is 0.17 and the right side tells us that the probability that we will win one dollar is 0.83 in other words the probability that we will win one dollar is much higher than the probability that we will lose one dollar and that makes it seem like it would be a good idea to accept the bet however even though there is a high probability that we will win the bet there is still a low probability that we will lose and no one likes to lose money so we say can we make this bet 100 times or is this just a one-time offer and our friend statsquatch says we can make this bet 100 times now if we make this bet 100 times we will probably win some and we will probably lose some and we can use this table to predict how much we will win and lose if we make this bet 100 times we can approximate how many times we will lose by multiplying the probability we will lose 0.17 by 100 and if we do the math we get 17. so that means we expect to lose about 17 times in 100 bets since we will lose one dollar each time we lose the bet we can estimate the total amount of money we will lose by multiplying the number of times we expect to lose by negative one so this whole term represents how much money we expect to lose in 100 bets and when we do the math we get negative 17 and that means we expect to lose about 17 dollars now since we can make the bet 100 times we can approximate how many times we will win by multiplying the probability we will win 0.83 by 100 when we do the math we see that we expect to win about 83 times since each time we win the bet we will win one dollar we can estimate how much money we will win by multiplying the number of times we expect to win by one so this whole term represents how much money we expect to win in 100 bets and when we do the math we see that we expect to win 83 dollars now that we have a term for the expected amount of money lost and a term for the expected amount of money won we can add the two terms together to find out the total of how much we expect to win or lose and when we do the math we see that we expect to gain approximately dollars after 100 bets however we can also calculate the average amount of money we will gain per bet by dividing everything by the number of bets 100 doing the math gives us 66 divided by 100 which is 0.66 so on average we expect to gain 66 cents every time we bet note even though i win or lose one dollar each time i bet on average i expect to gain 66 cents each time in statistics lingo 66 cents is the expected value for the bet bam using fancy statistics notation we could write e of bet equals 0.66 or if we wanted to make it look even more cryptic and more in line with what you might find in a textbook we could write e of x equals 0.66 where x represents the bet now let's talk about why i left this messy math in the middle of everything since we are multiplying each probability by the number of bets 100 and dividing by the number of bets 100 then all of the values that represent the number of bets 100 cancel out and we're left with the probability that someone in statland has heard of troll 2 times the outcome negative 1 plus the probability that someone has not heard of troll 2 times the outcome 1. and when we do the math we get the exact same thing we got before 0.66 thus the expected value represents what we would expect per bet if we made this bet a bunch of times note we can rewrite the expected value using fancy sigma notation using fancy sigma notation the expected value e of x is the sum of each outcome x times the probability of observing each outcome x so for the first term herd of troll 2 the outcome is negative 1 and the probability of observing the outcome is 0.17 so we multiply those values together then the sigma tells us to add that term to the term for not heard of troll 2 now the outcome is 1 and the probability of observing the outcome is 0.83 so either way we do the math we get 0.66 that means that if we can make this bet a bunch of times even though we will lose some of the time we should make money in the long run double bam now imagine statsquatch saying because it is relatively rare for someone in statland to have heard of troll 2 i will pay you 10 if the next person we meet has heard of troll 2 but if they have not you pay me one dollar will we win money or lose money if we can make this bet a bunch of times let's calculate the expected value to find out the outcome for when someone has heard of troll 2 is 10 because we will gain 10 dollars and the outcome for when someone has not heard of troll 2 is negative 1 because we will lose one dollar so these are the two outcomes now let's calculate the expected value the expected value is the sum of each outcome times its associated probability so let's start by plugging in the numbers for herd of troll two first the outcome x is ten and the probability of observing that outcome is 0.17 now we add the term for never heard of troll 2. the outcome x is negative 1 and the probability of observing that outcome is 0.83 now we just do the math and the expected value is 0.87 and that means we expect to gain on average 87 cents every time we make this bet and that means stat squatch is the worst gambler ever triple bam note in this stat quest we have only talked about how to calculate expected values for discrete events like whether or not someone has heard of troll 2. however we'll talk about how to calculate expected values for continuous events like how much time passes between text messages on your phone in another stat quest lastly if you watch this video hoping to learn exactly why we divide the sample variance by n minus 1 just know that this is the first of several steps to that answer and will get there someday soon now it's time for some shameless self-promotion if you want to review statistics and machine learning offline check out the statquest study guides at statquest.org there's something for everyone hooray we've made it to the end of another exciting stat quest if you like this stat quest and want to see more please subscribe and if you want to support statquest consider contributing to my patreon campaign becoming a channel member buying one or two of my original songs or a t-shirt or a hoodie or just donate the links are in the description below alright until next time quest on

Channel: StatQuest with Josh Starmer

Views: 23,823

Rating: 4.9783392 out of 5

Keywords: Josh Starmer, StatQuest, Machine Learning, Statistics, Data Science

Id: KLs_7b7SKi4

Length: 13min 39sec (819 seconds)

Published: Sun Mar 28 2021