What Expected Value Actually Means
Expected value is the average outcome of a bet if you could repeat it an infinite number of times. Not the outcome of one spin, one hand, or one session — the long-run mathematical average. It compresses every possible result of a wager into a single number that tells you whether the bet favours you or the house.
The formula is straightforward:
EV = Σ(probability × outcome)
That is: for every possible result, multiply its probability by its payout (or loss), then add them all together. The number you get is the expected value of the bet. Positive means the bet favours you over time. Negative means it favours the house. Zero means it is perfectly fair.
Nearly every bet you will encounter in a casino has a negative expected value. That is not an opinion. It is arithmetic.
Worked Example 1: A Fair Coin Flip
Start simple. You flip a fair coin. Heads, you win $10. Tails, you lose $10. Two outcomes, each with a 50% probability.
EV = (0.50 × +$10) + (0.50 × -$10) = $5 - $5 = $0
Expected value: zero. Neither side has an edge. This is a fair bet. You will not find it in any casino, because casinos do not offer fair bets. They are businesses. The house edge is their margin.
Worked Example 2: A Roulette Bet
You place $10 on a single number in American roulette. There are 38 pockets (1-36, plus 0 and 00). One pocket wins, 37 lose. The win pays 35:1.
EV = (1/38 × +$350) + (37/38 × -$10)
EV = $9.21 - $9.74 = -$0.53
For every $10 you bet on a single number, you lose an average of 53 cents. The house edge is 5.26%. This holds for nearly every bet on the American roulette table — the maths is the same whether you bet red, odd, a column, or a single number. The edge is structural, baked into the gap between true odds (37:1) and the payout (35:1).
Key Point
The house edge is not hidden. It is the mathematical difference between the true odds and the payout odds. In roulette, there are 38 pockets but the winning payout is calculated as though there are 36. Those two extra pockets — 0 and 00 — are the house edge, expressed in felt and chrome.
Worked Example 3: A Slot Spin
Slot machines publish a metric called Return to Player (RTP). An RTP of 96% means that, over a very large number of spins, the machine returns 96 cents for every dollar wagered. The house edge is 100% minus RTP — in this case, 4%.
You bet $1 per spin on a 96% RTP slot.
EV per spin = $1 × -0.04 = -$0.04
Four cents lost per spin, on average. That does not sound like much. But slots are fast. At 600 spins per hour — a typical pace — you are exposing $600 per hour to a 4% edge. That is $24 per hour in expected losses. The speed is the mechanism. Each individual spin costs almost nothing. The volume costs a great deal.
This is precisely why the bankroll calculator exists. It translates RTP and spin rate into session-level expected costs so you can see the real number before you sit down.
Prof. Boston says
I use a restaurant analogy in lectures. Imagine two identical steakhouses on the same street — same menu, same ambience, same steak. One charges $48 and the other charges $72. Nobody would walk into the expensive one by accident. But that is exactly what happens when a player chooses a 94% RTP slot over a 96% RTP slot for a four-hour session. The experience is nearly identical. The price difference is $48. The only reason people pay it is that casinos do not print the hourly cost on the machine.
The Big One: Expected Value of a Casino Bonus
This is where expected value becomes genuinely powerful — and where most players get it wrong. Casino bonuses look like free money. They are not. They are products with a price, and EV is the tool that reveals the price tag.
A standard offer: deposit $100, receive a $100 bonus, with a 35x wagering requirement on the bonus amount. You must wager 35 × $100 = $3,500 before you can withdraw anything.
Step 1: Calculate the total amount wagered
Wagering requirement × bonus amount = $3,500 in total bets.
Step 2: Calculate the expected house take
If you clear the wagering on a 96% RTP slot, the house edge is 4%. On $3,500 in wagers, the expected cost is:
$3,500 × 0.04 = $140
Step 3: Calculate the net expected value
You received $100 in bonus funds. You are expected to lose $140 while clearing the requirement. The net EV of the bonus is:
$100 - $140 = -$40
The bonus has a negative expected value of $40. You are, on average, paying $40 for the experience of wagering $3,500. The "free" money costs more to unlock than it is worth.
Key Point
The formula for bonus EV: Bonus Amount - (Wagering Requirement × Bonus Amount × House Edge). If the result is negative, the bonus costs you money in expectation. Most standard bonuses with 30x+ wagering on slots are negative EV. The higher the wagering multiplier, the worse the deal. A 50x requirement on the same bonus pushes the expected cost to $100 — the bonus is worth exactly nothing.
Not every bonus is negative EV. Low wagering requirements (10x-15x) on high RTP games can produce positive expected value. The bonus analysis page walks through how to identify those offers and why they are rare.
Negative EV Does Not Mean You Cannot Win
This is the part that people misunderstand, and it matters. Expected value describes the average over many repetitions. It does not describe any single session. You can sit down at a negative EV game and walk away a winner. People do it every day. That is not a flaw in the maths — it is a feature of variance.
In the short term, anything can happen. The distribution of outcomes around the expected value is wide, especially in high-volatility games like slots. A single session is a tiny sample. One player might win $500 in an hour on a game with a -4% edge. Another might lose $300. Both outcomes are consistent with the same expected value.
But — and this is the critical distinction — as the number of bets increases, the actual average result converges on the expected value. This is the law of large numbers, and it is not negotiable. Over hundreds of sessions, over thousands of hours, the house edge asserts itself with mathematical certainty. The casino is not gambling. It is running a business with a known margin. The variance that gives you hope in the short term is the same variance the casino has averaged away across millions of players.
Prof. Boston says
Here is a question I pose to every class: if a casino offered you a contract that said "pay us $96 per four-hour session and we will let you press a button with flashing lights," would you sign it? Most students laugh and say no. Then I show them that the contract already exists — it is called a $1-per-spin slot at 96% RTP played at normal speed. The only difference is that variance obscures the price. Some sessions cost you $20. Some cost you $200. But the average, across enough sessions, converges on that contract price. Variance is not your friend. It is the casino's camouflage.
Why EV Is the Framework Behind Everything on This Site
Every tool, every analysis, every recommendation on this site starts with expected value. The bankroll calculator converts RTP into session-level EV. The bonus evaluator uses EV to separate genuine offers from marketing theatre. The session management framework exists because EV tells you the cost of continued play, which means it also tells you when the cost exceeds the value of the entertainment.
The psychology articles on this site explain why your brain makes it hard to act on EV — loss aversion makes you chase losses, near-misses make you overestimate how close you are to winning, and variable reward schedules make you resistant to stopping. All of these biases operate by distorting your perception of expected value. They make negative EV feel positive.
Understanding EV does not make you immune to those biases. But it gives you a reference point — a number you calculated before the session started, before the emotional machinery kicked in. When the machine says keep going and your pre-session maths says the cost is $24/hour, you have a conflict between feeling and arithmetic. The arithmetic is always right.
That is not a small thing. It is the foundation of every informed gambling decision. Start here. Build from here. The rest of the site exists to help you act on what this page teaches.