Expected value in gambling ·
Further reading — Module 04
What Is a Bet Really Worth?
A bet can win and still be a bad bet. It can lose and still have been a perfectly fair one. That sounds contradictory until you separate two things: how likely an outcome is to happen, and what you are paid when it does. In gambling, expected value brings those two things together. It asks a simple question: does the payout fairly reflect the probability?
01 — Start with the probability
A fair coin and a fair payout
Imagine a perfectly fair coin. Heads has a 50% chance of appearing. Tails has a 50% chance of appearing. Nothing favours either side.
Now imagine risking £10 on the result. A correct call wins £10 and an incorrect call loses £10.
It makes no difference whether you choose heads or tails. The probability is 50/50, and the payout matches it.
The bet is mathematically balanced.
Fair payout — no house edge
The probability and the payout are in balance.
The setup
A fair coin has two outcomes.
Heads has a 50% chance.
Tails has a 50% chance.
You risk £10.
A correct call wins £10.
An incorrect call loses £10.
The coin is fair.
The payout matches the probability.
Probability × payout
0.50 × £10 = +£5.00
0.50 × −£10 = −£5.00
Expected value
+£5.00 + −£5.00 = £0.00
02 — Now change the payout
The coin stays the same
Now change only the payout.
You still risk £10, but a correct call now wins only £9.
It does not matter which side of the coin you choose. The probability is still 50/50, but a win now gives you less than a loss takes away.
The coin is still fair. The payout is not.
Unfair payout — house edge
The probability is unchanged, but the payout is smaller.
The setup
The same fair coin still has two outcomes.
Heads has a 50% chance.
Tails has a 50% chance.
You risk £10.
A correct call now wins £9.
An incorrect call still loses £10.
The coin is still fair.
The payout no longer matches the probability.
Probability × payout
0.50 × £9 = +£4.50
0.50 × −£10 = −£5.00
Expected value
+£4.50 + −£5.00 = −£0.50
03 — The payout implies a different probability
The mismatch is the disadvantage
This is the important step. A 50/50 outcome needs a payout that reflects a 50% chance if the bet is going to be fair.
When £10 can win £10, it does.
But when £10 can win only £9, winning half the time is no longer enough to break even. You would need to win slightly more often than the coin tends towards.
The underlying probability is still 50%. But the £9 payout requires a winning rate of about 52.6% to break even.
That difference is the mathematical disadvantage. You do not need to predict the next toss to see it. The probability and the payout simply no longer match.
The coin
- Underlying probability of winning: 50%
The payout
- Risk £10 to win £9
- Winning rate required to break even: 52.6%
Actual probability: 50% · Required probability: 52.6% — the gap creates negative expected value.
04 — Where expected value enters
A value for the mismatch
Expected value puts a financial value on the difference we have just seen.
With the £9 payout, the winning half contributes £4.50 on average while the losing half costs £5.00.
The difference is −50p per bet.
That does not mean every bet loses 50p. An individual bet still either wins £9 or loses £10.
−50p is the average mathematical value of making that same decision repeatedly.
That is expected value.
05 — Repetition reveals the difference
Randomness and expectation
This connects directly with When Luck Disguises Reality. One toss can produce either result. A short sequence can produce almost any pattern.
But repeating the same proposition means repeatedly exposing yourself to the same relationship between probability and payout.
Short-term randomness determines which individual bets win. Expected value describes the mathematical advantage or disadvantage built into those bets.
That is why the outcome of an individual wager can be unpredictable while the underlying value of the proposition can still be known.
The view from behind the tables
“From behind the tables, individual results could appear to tell completely different stories. One player could leave ahead while another lost heavily. Either could happen on any given night.
But the casino did not need to predict those individual outcomes. What mattered was the relationship underneath them — the probability of each result and the payout attached to it.
The next result was uncertain. The mathematics of the proposition was already there.
The mantra was simple: the more we spin, the more we win.”
Final thought
Does the payout match the probability of winning it?
The payout odds are never hidden.
They are clearly displayed in the rules and paytable plaques on every table.
But once you understand expected value, you may start to see them differently.
The next time a winning bet is paid out, will you see what you got?
Or what you should have got?
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