If someone told me when I was a kid that studying math could make me rich, I would’ve subtracted that paper airplane from my hand and added some serious homework time into the equation. But no one did, really, and as far as I was concerned back then – math nerds weren’t cool! But like most of the other decisions I made in my younger days, this one came back to bite me in the ass soon enough. Expected Value comes straight outa the math books of Hustle College, and you’re about to find out why.
So How Do I Calculate Expected Value?
Hold on to your panties. We’ll get to that in a second, but before we do let’s talk about why this number is the reason you could be homeless if you don’t play your cards right (see what I did there?).
In 2015, a word association study involving approximately five thousand participants conducted by professors from the Las Vegas University’s Hustlenomics Department, determined that over 90% responded indicating the first word that comes to mind when they hear the word ‘Casino’ is – ‘Luck’.
Alright, I’m bluffing. There is no such University.
The point I’m making is not many people would immediately think of ‘Mathematics’ when they hear ‘Casino’. And actually, that’s really all there is to it.
You Can Bet On It
When you place a bet, everything boils down to Expected Value. At least it should, because that’s how Casinos make money from you. So what exactly is Expected Value? Simple:
The expectation value of a function in a variable is denoted or . For a single discrete variable, it is defined by
where is the probability density function.
Oh Chillax. Obviously I’m going to explain. Expected Value is actually a pretty simple concept to grasp. All it does is use probability theory to break down your ‘average’ win/lose to give you a mathematical determination of exactly how much you’re making or losing on that bet (ie. your win/lose rate in the long run). So if your bet delivers positive Expected Value, you will make a profit in the long run and theoretically it is a good bet – and vice versa. See the example below and it should make perfect sense.
Here’s An Example Of Expected Value Calculation
Think of someone you know who you consider to be a complete moron. Got it? Ok Good. So one day, you and this idiot friend decide to wager on coin flips. Turns out your friend, being such a dumb ass, agrees to pay you $2 every time the coin flips Heads while you only need to pay him $1 if it lands on Tails. Common sense will tell you to take this bet, and so will Expected Value:
The odds of winning/losing are 50/50. If you chart the results out over a decent sample size of flips, the results will no doubt reflect that. Assuming this bet pays $2 to $1 – the Expected Value will be neutral (0). That means if you bet $1 and receive $2 (your original bet of $1 + a $1 win) when you predict the correct outcome – you will not make any money in the long run simply because 50% of the time you will lose your original bet of $1 while the other 50% of the time you will be returned $2. I hope I haven’t lost you. Read it again if you need to but make sure you totally get this, because it’s important.
Now let’s go back to the fun little game with your friend. Again, the odds of winning/losing are 50/50. However, this time round, each time you place a bet you pay $1 when you lose but receive $3 (your original bet of $1 + a $2 win) when you win. So how do you calculate the Expected Value? You simply multiply the probability and value for each outcome and sum them up like this:
[(Probability of you winning) x (Amount you will win)] + [(Probability of you losing) x (Amount you will lose)]
Which in this example will be:
Probability of winning = 0.5
Probability of losing = 0.5
Amount you will win = $3
Amount you will lose = $1
(0.5 x 3) + [0.5 x (-1)] = 1
So because the end result is a positive number (+1), this bet has positive Expected Value (or as the cool casino nerds usually write it, “+EV”).
“No wait you idiot…I started playing and it was Tails 3 times in a row so I lost $3!”
Sid: Shut up you fool! Mathematically speaking, you actually just made $3!
Variance: Expected Value’s Evil Stepsister
What this essentially means is that over the long run, regardless of the outcome of the current result you will be making $1 on every single bet. This result would become far more apparent when you play a large number of times – so let’s say if you play 1,000 times, you would roughly be winning $1,000. I’m not going to chart this out 1,000 times for you but let me show you how this may work using a fabricated ‘perfect scenario’ ten coin tosses:
|Toss 1||Tails||You lost $1|
|Toss 2||Tails||You lost $1|
|Toss 3||Tails||You lost $1|
|Toss 4||Heads||You won $3|
|Toss 5||Heads||You won $3|
|Toss 6||Tails||You lost $1|
|Toss 7||Heads||You won $3|
|Toss 8||Tails||You lost $1|
|Toss 9||Heads||You won $3|
|Toss 10||Heads||You won $3|
|TOTAL||You won $10|
In this set of 10 coin tosses that occurred between you and your friend, the odds of winning and losing are reflected perfectly – 5 wins and 5 losses. So when you look at the big picture, every bet makes you $1 (10 bets, $10). Get it?!
So in the short-run, if you play just 3 coin tosses, ‘luck’ can indeed become a big factor in determining your ‘instant results’ but over the long-run, if you make a +EV bet you will indeed be seeing a profit. Of course just 10 coin tosses is not a good enough sample size but if you run this thousands and thousands of times you will see that indeed it averages out to you making $1/bet. We’ve come up with an App called Baccarat Pro that will make doing this a breeze, so do check it out if you enjoy testing strategies!
“Ok thank you Sid. You’re my hero!”
Sid: I know, you’re welcome. Sorry I called you a fool earlier.
Once you fully grasp the concept of Expected Value you will become a much smarter punter, I assure you. Of course the example above is an extremely simple one and as we go through more articles and strategies we will look at more complex cases – but understanding this example is a crucial first step to building your foundation. So congrats if you get it, and read it again if you don’t.
Every single bet you make at a Casino has a negative expected value (-EV), just in case it isn’t obvious enough by now. So just like the coin toss example earlier, Casinos know that as long as the sample size is big enough they will never lose money. You may stroll in one day and win 3 bets in a row (like the first 3 coin tosses in the example), but there’s always someone else there who will be losing to make up for it, just like eventually if you play long enough you will too. Big Picture! Remember?
Hey! Take a gamble by connecting with us on social media. Odds are you’ll love it.
References: Weisstein, Eric W. “Expectation Value.”