Expected Value (EV) Explained


Recently, a friend of mine told me he had a full-proof method for playing at the casino.  Naturally, as a fellow gambler and poker player, I was curious. He told me he had a betting system for Roulette. Immediately, I became skeptical. By no means am I a professional gambler (poker isn’t gambling), but I’m aware that Roulette not only holds one of the highest casino edges (aside from slots), but I’m also aware that betting systems (i.e. the Martingale) are absolutely terrible (it involves doubling your bet after each losing round, revolving around the assumption that you’ll eventually win back your original loss; plus some). The problem with games such as Roulette, and with systems such as the Martingale, are their short-sightedness. A lot of people get excited when they bet 1/3 of the numbers on the Roulette wheel, hit a number and end up ahead, or even worse, considerably ahead. A lot of individuals also assume that with a system such as the Martingale, how many hands of Blackjack can I really lose in a row? The truth is, MANY!!! We must consider the long term benefits of such plans. But I digress.

My friend told me that he consistently bets 2 out of the 3 twelve blocked number groups (1st 12, 2nd 12, 3rd 12) since the percentages for those 2 bets together are equivalent to 66.6% (33.3% x 2) of the numbers on the board. I told him, GREAT! Then I quickly asked, “Well what’s your EV?” That’s when I received the dreaded look of absolute bewilderment. EV refers to the term “expected value.” EV can be defined as the amount, on average, that you should gain (or lose) from making a particular play, or in my friend’s example, a bet.

I quickly told my buddy that casinos rarely offer bets in which they’re an underdog, and that I would quickly disperse of any hope he still held that he could make money off of this betting scheme in the long run. Still following? Good. Let’s look into this in further detail.

A roulette wheel usually contains 38 numbers. Numbers 1 through 36, with the addition of 0 and 00 (some wheels only contain 0, which gives players a slightly better advantage). As stated above, “Friend” said he bets 2 sets of 12 numbers, which is 24. This leaves 14 left. Already, “Friend” made a fatal error of miscalculating his equity, as 24/38 equals 63%, not 66.6%. For the sake of simplicity, let’s say the table minimum at this particular Roulette game was $10 a bet. Now, when paid on one set of 12 numbers, the casino odds you are paid are 2 to 1. Meaning, if you bet $10, you would be paid an extra $20, or have $30 total. We need this information to calculate our expected value. So in order to solve an equation, we need two sides. Those two sides are what we WIN, and what we LOSE. So when “Friend” bets $20 (2 $10 bets), if he wins, he’ll lose 1 bet, and win on the other. This nets us $10 in PROFIT ($20 we won – $10 we loss). If “Friend” loses, well, that’s easy. He loses both, or $20. Now to compare our expected value, we use our percentages. Our percentage of winning the Roulette bet was 63%, while our percentage of losing is 37%. So here is the formula:

($10 x .63) + (-$20 x .37) = -$1.10 EV

So we’ve multiplied our win $ against our win %, and our lose $ against our loss %. Is the result surprising? Not really! On average we’re expected to lose $1.10, even though the odds (at least percentage wise) are in our favor. Sure, like most casino patrons we may feel the utmost gratification in our short-term wins, and may feel invincible. But don’t be short-sighted. Since you’re losing double the amount you’re winning, you need your win percentage ratio to be greater than 2 to 1 to make money (when I told my friend this he was sincerely dissapointed). Make sense? I’ll explain further.

In poker, there are certain times when you need to be able to calculate your profitability in certain situations. Ask yourself, “If I bet here, how often will I take this pot down uncontested?” The answer to this question differs from opponent to opponent. Let’s look at a poker example.

You’re playing $5/$10NL. Everyone folds to you at the button with 4s5s. You’ve been playing loose-agressive, but within reason. You raise to $35. The small blind re-raises to $80, and the BB folds. You call to a flop of 2h3h8d. He bets $165, (almost pot! WHAT!) you call. Turn 5h. He checks, you check. River Qc. He checks. He’s a tight-agressive regular, who normally doesn’t get too out of line. When guessing his range, and his line, we could reasonably assume he would play hands like 99-JJ this way. Probably, A-K and a slight chance of A-Q, although he probably bets the river. If he’s feeling froggy, maybe A-J, A-10, or suited aces. Maybe even K-K or A-A, depending on your range and table image. Most likely he holds something that can’t raise a bet here, but has showdown value. We’ll include all the hands above, but also J-Q, K-Q, 8-9, and Q-10 type hands. It’s unlikely he has a flush here, since he is aggressive, and he misses value from overpairs that called pre-flop. Let’s say you estimate that your opponent will call 50% of the time, and fold 50%. Your hand essentially has no showdown value on the river. You bet $375 into your opponent, who now has to make a tough decision on whether or not to call, or fold. So let’s examine the EV of this particular hand, using our equation.

($500 x .5) + (-$375 x .5) = +$62.50!!

If you take the pot down uncontested, you’re going to win $500. Simple. If your opponent decides he wants to make a hero call, so be it. You lose the $375 you bet. But the great news is, is that after doing the math, this play becomes a profitable move to make over the long run, as you average a win of $62.50 against this particular opponent!!! Of course, the key is to allow your percentage estimates to be as accurate as possible (HEM or Poker Tracker may help in that situation.)

Calculating EV is a tool that all great professionals use to calculate their chance of winning a hand. More often than not, you’ll need this information to calculate your equity on drawing hands, in order to find out whether or not it becomes profitable to call a bet with your draw given the right pot odds (for more info on pot odds & equity, search the plethora of sites available on the internet.)

Usually the next question I’m often asked is, “So I can’t calculate the EV of checking?” In a short answer, no, at least not mathematically. However, what happens if “you check, your opponent bets, and then you check raise?” Or “both of you check, and you make the nuts or a better hand than his on the turn?” Even better, what happens if you simply “check, he bets, and then you call?” How often do you believe you may win the pot in these situations? That can all be calculated. It’s very important to realize that most moves (if not every) has an expected value. Don’t find out the hard way like Mr. Roulette man. Use this information to your advantage, and you’re on your way to thinking and playing poker like a professional.

Good luck.

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