Expected value ·
Further reading — Module 04
What Is a Bet Really Worth?
A bet can win and still be a bad bet. Two numbers decide that: the chance of the outcome, and the price you are paid for it.
01 — Two numbers
Every bet contains a chance and a price
Imagine a fair coin and a £10 bet on heads. Heads occurs half the time. If the bet is genuinely fair, a win pays £10 and a loss costs £10. Probability and payout are in balance.
The chance
- "Heads — 50%."
- "Tails — 50%."
- "Nothing about the coin changes."
This is the part most people focus on: will it come in?
The price
- "Risk £10 to win £10."
- "Risk £10 to win £9."
- "Risk £11 to win £10."
This is the part the house sets — and it is where the advantage is stored.
02 — Change nothing but the payout
The price changes the bet
Leave the coin exactly as it is. Heads is still 50%. Now suppose a win produces only £9 profit on a £10 stake. You still win half the time — the winning half simply no longer covers the losing half.
The randomness did not change. The probability did not change. The payout did.
03 — The gap is the whole story
True probability against implied probability
There is the probability of the event itself, and there is the probability implied by the price on offer. When those two numbers match, the proposition can be mathematically fair. When the payout requires an outcome to occur more often than it really does, the person taking the bet is accepting a mathematical disadvantage.
Expected value puts a figure on that disadvantage. It answers a different question from “Will I win?” It asks: if this same decision were made again and again, what would each one be worth on average?
04 — Results are not the proposition
You can still win a bad bet
Negative expected value does not mean the next bet will lose, or that the session will. It describes the proposition, not the outcome. A player can make a negative-EV decision and win; another can make a positive-EV decision and lose.
The result tells you what happened. Expected value tells you what the decision was worth before the result was known — and that is the distinction the whole subject rests on.
05 — Repetition exposes the difference
Randomness sets the route, expectation sets the direction
One result can hide almost anything. A run of results can hide quite a lot. But repeating the same proposition gives its built-in advantage or disadvantage more chances to show itself — the same mathematics traced in When Luck Disguises Reality.
That is why a business taking millions of wagers never needs to know which customer will win. It only needs the relationship between probability and payout to stay in its favour while the decisions repeat.
The view from behind the tables
“From behind the tables, individual results could look completely contradictory. One player left hundreds of pounds ahead while another lost just as much. Neither result told you whether the underlying proposition had changed.
The casino never needed to know who would win the next hand, spin or wager. It needed something far more reliable: a relationship between probability and payout that stayed in its favour while the decisions repeated.
The players experienced the results. The business relied on the expectation.”
Final thought
The next time a wager advertises what you could win, look for the other number hiding behind it. How likely is that outcome, really — and does the payout honestly reflect it? The difference between those two answers is what the bet is actually worth.
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