How to work out the expected value of a bet
Multiply the true probability of winning by the decimal odds, subtract 1, and multiply by the stake. For an even chance priced at 1.90 with KES 100 staked: 100 x (0.50 x 1.90 - 1) = -5. On average that bet costs KES 5, because the price includes a margin.
The idea in one paragraph
Expected value (EV) is the average result of a bet if the same bet could be repeated a very large number of times. It is not what will happen next. It is the figure that results settle around in the long run, and for bets priced by a bookmaker it is below zero. To find the margin that causes this, start with the guide on calculating the bookmaker's margin.
The formula
Here p is the true probability of winning, D is the decimal odds and the profit on a win is stake × (D − 1). The two lines are the same sum; the second is quicker.
The sign tells the story. If p × D is below 1, the EV is negative. Since the price's own implied probability is 1 ÷ D, the EV is negative whenever the true probability is lower than the implied one, and the margin makes sure that it normally is.
Step by step
- Write the price as decimal odds. The odds converter handles fractional and American prices.
- Decide what probability to use for
p. For a coin toss or a dice roll it is known. For a match it is an estimate, and that is the weak point of the whole exercise. - Multiply
pbyDand subtract 1. This is the EV per shilling staked. - Multiply by the stake to get shillings.
Illustrative example
An event has two equally likely outcomes, so p = 0.50. A fair price would be 2.00. The price offered is 1.90 and the stake is KES 100.
EV = 100 × (0.50 × 1.90 − 1) = 100 × (−0.05) = −KES 5. Half the time the bet makes KES 90, half the time it loses KES 100, and the average of those two results is a loss of KES 5.
How the result moves with the true probability
The table keeps the price at 1.90 and the stake at KES 100, and changes only the true probability of winning.
| True probability p | p × D | EV per KES 100 staked | Reading |
|---|---|---|---|
| 40.00% | 0.760 | −KES 24.00 | The outcome is less likely than the price implies |
| 45.00% | 0.855 | −KES 14.50 | Still well below break-even |
| 50.00% | 0.950 | −KES 5.00 | The even-chance case: the cost is the margin |
| 52.63% | 1.000 | KES 0.00 | Break-even: p equals the implied probability, 1 ÷ 1.90 |
| 55.00% | 1.045 | +KES 4.50 | Positive only if the true chance really is this high |
The last row is arithmetic, not an opportunity. To land in it you would need to know the true probability better than the price does, and no formula on this site, or anywhere else, supplies that knowledge.
The problem with p
For a casino game the probabilities come from the rules of the game, and regulation 20(2) of the Gambling Control (Conduct of Gambling Operations) Regulations, 2026 requires a licensee to clearly display rules of games, odds, house edge and average return to the player. Regulation 45 adds that the theoretical return to player percentage for each online game shall be disclosed. In those games the EV per shilling is simply the return to player minus 1, and it is negative by design.
For sports there is no rulebook probability. Any p you choose is an opinion. It is easy to be too confident about a team you follow, and a small error is enough to flip the sign: in the table, the gap between a 5 shilling cost and break-even is under three percentage points of probability.
The cautious reading is therefore this. Use the margin-free probability of the market as your p, and the EV of every bet in that market comes out negative, by roughly the size of the margin.
Why one bet tells you nothing and many bets tell you a lot
A single bet never returns its expected value. The KES 100 bet above ends at +90 or −100, never at −5. This is why a short run of results says very little, and why a few early wins are so misleading.
Over many bets the picture changes. The spread of results around the average grows only with the square root of the number of bets, while the expected cost grows with the number itself.
Illustrative example
Take 200 bets of KES 100 each on even chances at 1.90. Total staked: KES 20,000. Expected result: 200 × (−5) = −KES 1,000.
The typical spread (one standard deviation) for one such bet is 100 × 1.90 × √(0.5 × 0.5) = KES 95. For 200 independent bets it is 95 × √200, about KES 1,344. So a result somewhere between roughly −KES 2,344 and +KES 344 is ordinary, and most of that range is a loss.
At 2,000 bets the expected result is −KES 10,000 and the typical spread is about KES 4,249. Ending ahead would now need a result more than two standard deviations above the average, which is rare.
This is the practical meaning of "the house always wins". It is not that every bet loses. It is that the average cost keeps adding up while luck, in proportion, fades.
What EV does not justify
- Raising stakes after losses. Each new bet has its own negative EV. Bigger stakes make the expected cost bigger, not smaller.
- Borrowing to bet. A negative-EV activity cannot repay a loan on average. Section 74(1)(a) of the Gambling Control Act, 2025 also bars a licensee from providing credit to a player or on a player's account.
- Treating betting as earnings. An activity with a negative average result is a cost, like any other paid entertainment.
How to use the number sensibly
EV is best used as a price tag. Multiply the amount you expect to stake in a month by the average cost per shilling, and you have the expected price of that month's betting. The session budget planner does this sum and divides a monthly ceiling into sessions.
Then compare the price tag with what the pastime is worth to you, and remember that the actual bill can be the whole amount staked. Combining selections makes the price tag grow faster, as the guide on accumulators and margin shows.
Questions and answers
What does a negative expected value mean?
It means that the bet costs money on average. Any single bet still either wins or loses in full; the expected value describes the average over a very large number of identical bets.
Can the expected value of a bet be positive?
Only if the true probability is higher than the one implied by the price. Nobody can observe the true probability of a sporting event, and prices already include a margin, so a positive figure on paper usually reflects an optimistic estimate, not an advantage.
Is expected value the same as the bookmaker's margin?
They are two views of one thing. The margin is measured across all outcomes of a market; expected value is measured for one bet. If the margin is spread evenly, the expected value per shilling staked is the payback minus 1.
Does a staking system change the expected value?
No. Changing the size or order of stakes changes how results are distributed, not their average. A sum of bets that each have negative expected value has negative expected value.
If I keep betting long enough, will my results even out?
They move toward the expected value, which is negative, not toward zero. The longer the series, the more certain it becomes that the total result is a loss close to the average cost.
Sources
- Gambling Control (Conduct of Gambling Operations) Regulations, 2026, rr.20(2), 23, 45, accessed 2026-10-04
- Gambling Control Act, 2025 (No. 14 of 2025), ss.2, 74, accessed 2026-10-04