30.04.2024 The Mathematics of the Casino | What people get wrong about gambling

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This video was sponsored by notion. let's say alice and bob go into the. casino to play some roulette with only. one objective in mind. win 100. but they both have different strategies. and different starting amounts. bob only has 100 to begin so he's going. to keep it short he will bet 100 on red. once if he loses he's done he lost his. 100 and if he wins he gets 100 red pays. 11 so he walks away because he hit his. goal he's done alice on the other hand. comes in with ten thousand dollars but. she's only going to bet one dollar on. red over and over until she's either up. a hundred dollars or she loses. everything she won't stop until one of. those happens. and remember red only pays out whatever. you bet so she needs a hundred more wins. than losses. before everything is gone. and note there are 18 out of 38 spots.

That are red so that's roughly a 47.4. percent chance any individual spin lands. red. okay now the question is who has the. better chance of walking out of this. casino having made one hundred dollars. and before you attempt to prove this on. your own just note the proof itself is. not obvious or meant for a video like. this but i will provide a link below for. more detail. okay here we go since there are 18 out. of 38 spots that are red that means bob. simply has a roughly 47.4 percent chance. of getting his 100. and a 52.6 percent chance of losing. everything he's only doing one bet so. the math is easy. alice on the other hand ray for this. has a point zero zero two six five. percent chance of winning that one. hundred as in it is almost certain she's. losing all ten thousand before she makes. just one hundred.

If you didn't get that it's okay you got. one more try because here comes the. richest man in the universe carlos. he is going to do the exact same thing. as alice he's going to bet one dollar at. a time until he wins 100 or loses. everything. but. he's starting with a hundred trillion. dollars. so he's just got to get a hundred more. wins than losses before he gets a. hundred trillion losses over wins. so where the odds he walks away reaching. his 100 goal or how much more likely is. he to win compared to alice since i mean. he's doing the same thing but has more. opportunity to win. well guess what he has a .00265. percent chance of winning 100. again it is almost certain he will lose. 100 trillion before he makes 100. he is. no better off than alice. now if that bugs some of you you're. right these numbers are not the same.

Carlos has to have better odds and he. does. but the difference is so small i had to. use a high precision calculator just to. find it and here it is. so they basically have the same chance. of winning one hundred dollars. now the equation to figure this out that. tells you the odds of going up some. amount given your starting bankroll is. this here. and again how it is derived is not. obvious not for this video but i'll put. a link to an mit lecture down below. where the professor goes through the. proof. so for the people who do this strategy. in the casino but the same amount over. and over this equation tells you. everything you need to know. here let's make a real scenario where. you have 20 units to play with pretty. common as in like you're playing a 10. roulette table and you have a total of. 200 to play with or it could be a 100.

Table starting with 2 000 whatever you. simply have 20 times the minimum allowed. bet. now if your strategy is i will bet 10 on. red or black over and over until i make. m dollars or until i've lost all 200. then here's the probability e of. succeeding given different values of m. so you're actually more than likely to. win. if you tell yourself hey i'm going to. stop once i'm up 50. or anything less. so the important thing to note here is. that how you play absolutely affects. what your odds are of walking out that. casino with more money than you started. with a common phrase you hear is the. house always wins but that's not really. true yes it's true in the long run. casinos will make money but anyone can. walk into the casino and have a higher. chance of making money than losing money. in fact you can make the probability of.

Winning money really anything you want. between zero and a hundred percent not. inclusive sorry it can't be a hundred. but if you want a 99 chance of walking. out with a hundred or even a thousand. dollars you can make that happen the. thing is the the higher the probability. the more money you stand to risk. in general i mean we saw that wasn't. even the case with uh. alice and bob but in general if you want. a 99 chance of walking out of that. casino with a thousand dollars let's say. then you need a lot more than a thousand. dollars so you stand to risk more but. again you can make the probability. whatever you want for the most part. but within this fact is where the. casinos make money because that risk. always outweighs what you stand to make. in the long run. if everyone watching this video went to. a casino and tried this strategy let's.

Say to make thirty dollars then walk out. if you succeed. then most of you over seventy percent. are going to win thirty dollars. but the thirty percent of you who lose. will lose two hundred dollars over six. times more. and that total loss looking at. everyone's losses doesn't just cancel. out the total wins it exceeds it add up. all the wins and losses and the losses. will always be greater that's why the. house wins in the long run when looking. at everyone individually not necessarily. the case. so again the lesson here is how you play. and what games you play absolutely has. an effect on your odds of walking out. that casino with more money than you. started with or less and how much but. there is one thing in the casino one. mathematical concept that you cannot. change you cannot outrun you cannot. avoid this unless you're cheating but in.

General you cannot change this i'm going. to explain what that is after thanking. the sponsor of this video notion. notion is an amazing productivity tool. that you can use to stay organized. manage tasks take notes and much more. whether just for yourself or for any. kind of team so whether you're wanting. to be more productive at school or. you're running a company notion has. everything you'll need you need to make. a weekly agenda or an entire calendar. notion makes it real simple to add and. check off events and projects as needed. want to just take notes for class just. open a new page and start typing or if. you're in a business meeting all notes. can be taken and saved in notion so. everyone is sourcing from the same place. you can start conversations and leave. comments on collaborative projects so no.

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So if you'd like to save yourself time. and increase your productivity with. notion then just click the link in the. description below to sign up today. okay now getting back to the casino the. mathematical thing you cannot avoid or. change unless you're cheating is your. expected losses otherwise known as the. house edge. now the expected value or weighted. average can be thought of as what your. running average will approach after many. many samples of something. like the expected value of a single fair. die is 3.5 because yes it's the average. of the numbers but also if you rolled. that die over and over keeping track of. the running average. well it's going to bounce around in an. almost unpredictable way but it will for. sure approach 3.5 over time. so thinking in terms of expected values. or the house edge every single casino.

Game can be simplified to a coin toss. game. let's say we have a game where you flip. a fair coin and if it lands heads you. get a dollar if it lands tails you lose. a dollar you give it to quote the house. the casino whatever. this would be considered a completely. fair game as in no one has an edge and. your expected loss is zero. if you played this game over and over. and kept track of your winnings versus. number of tosses that y value would just. randomly bounce around zero sometimes. you're up sometimes down but the best. fit line you can think of would have a. slope of zero. however if we change the game where now. if it's heads you win 90 cents and tails. you lose a dollar this is now a biased. game where the house has an edge. that edge is 5. which is calculated from the fact that. half the time you win 90 cents and half.

The time you lose a dollar so that comes. out to an expected loss of 5 cents per. bet which is a loss of 5. of the dollar that you stand to lose. your bet essentially. so this game has a house edge of five. percent and if we played over and over. then we can expect our net profit over. time to look something like this. while chaotic it would dance around. instead of a line with no slope one that. looks like this. and if we kept playing and zooming out. the actual profit will look more and. more like that best fit line. that line has the equation y equals. negative point zero five x there we can. see that five percent number and we can. use this to make predictions that are. more accurate as a percentage as time. goes on. after you've played through 800 tosses. basically wagering 800 dollars in total. i expect you'll have lost about forty.

Dollars five percent of eight hundred. in this simulation i ran the actual loss. was forty six dollars and seventy cents. so pretty close. but that expected loss percentage five. percent tells us roughly what's going to. happen in the long run. if there was another game offered where. heads means you win 97 cents and tails. you still lose the dollar the graph. would look like this. here you can see the best fit line of. now y equals negative point zero one. five x from the house edge now being one. point five percent. after eight hundred bets just take one. point five percent of that and i assume. you have lost twelve dollars which is. close to the fifteen dollars and ninety. four cents lost in the simulation. won't always be that accurate as you can. obviously see but the point is over time. your actual losses will be fairly.

Predictable. now if you were forced to play one of. these two games which one would you. choose. well unless you're not a fan of money. should be pretty obvious that this one. is more in your favor all things equal. except you win more when it's heads. but there are several people who have. and will choose this game over the other. one well sort of without knowing it. because this right here. is pretty much roulette well this game. is craps. the coin flip game and the respective. casino games you see here are just about. identical in terms of the expected. losses or the house edge to be exact. double zero roulette has a house edge of. 5.26. no matter where you bet red black even. odd or individual numbers compared to. the five percent house edge of that. respective coin flip game. craps is a house edge of one point four.

One percent when you're betting on the. pass line compared to the one point five. percent of the coin flip game above. but just like before we can accurately. predict our future with these numbers at. least in the long run the math is no. different if you're betting 10 at the. roulette table and play for 500 spins. you'll have wagered five thousand. dollars in total house edge is five. point two six percent so just multiply. that by five thousand and you can expect. to have lost two hundred and sixty three. dollars. and if you did the same thing with craps. ten dollars per bet for 500 bets same. thing with five thousand dollars wagered. in total and at a house edge of one. point four one percent you can expect to. lose seventy dollars and fifty cents. and you can do this for any game when. you look at the house edge for anything.

In the casino it's simply telling you. you're expected to lose that percent of. every bet. multiply that percentage by how much. money you've bet in total and over time. that prediction is going to be more and. more accurate. and you might be asking what about. something like roulette for example. where you bet on an individual number. and you can get 35 times your bet if. that hits whereas the coin flip game. only had a oneto-one payout well it. doesn't matter there are 38 numbers in. total so you have a 1 out of 38 chance. of winning 35 times your bet and a 37. out of 38 chance of losing one of or one. times the original bet. do the math and look at that there's. still a 5.26 house edge. so if you're only betting one dollar on. red i bet only one dollar on 21 to go. for that 35 times payout and then. someone else plays the coin flip game.

With these rules. after hundreds or maybe thousands of. rounds. each person is going to have lost just. about the same amount of money because. through the lens of expected values. these are all the exact same game. and i know people go to the casino often. just for fun and roulette is obviously. much easier than craps or blackjack but. if you are genuinely looking to keep. your losses to a minimum or wins to a. maximum and you're debating between like. roulette craps baccarat or blackjack. just ask yourself which of these coin. flip games would you rather play because. it's all the same in the long run. and yes as you can see blackjack is the. best game to play in the casino if you. know basic strategy and have favorable. rules like blackjack pays 32 or you're. counting. and since i love playing craps i will. mention for any crafts players who like.

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