Expected value:
the average you would get if you could repeat a choice many times
A single decision rarely lands on the average. Still, knowing the weighted average is the cleanest way to compare options that carry different risks.
Calcylator Editorial Team
Updated · 5 min read
An average weighted by chance
Suppose you are choosing between two options, each of which could end well or badly. Comparing best cases or worst cases tells only part of the story. Expected value squeezes the whole spread of possibilities into one number by giving more weight to likely outcomes and less to unlikely ones.
It is the same idea as a weighted average. If a shop earns ₹40,000 in a strong month, ₹15,000 in a normal one and loses ₹12,000 in a weak one, the average month depends on how often each type happens. A strong month 1 in 10 times counts for much less than one that shows up 3 times in 10.
The term is used in probability, insurance, project appraisal, games and machine learning, always with the same meaning: what you would average per attempt if you could repeat the same situation a very large number of times.
The formula
- xᵢ:
- the value of outcome i (a gain, loss, score, price)
- pᵢ:
- the probability of outcome i
- Σ:
- add up across every possible outcome
- List every possible outcome and attach a value, using negatives for losses.
- Assign each a probability; check that they total 100%.
- Multiply each value by its probability.
- Add the products.
If the probabilities do not add up to 1, either an outcome is missing or a probability is mis-stated. This check is the most common source of errors with hand calculations.
Worked example: stocking a seasonal item
A shopkeeper can stock a festival item. From past years she estimates three outcomes: strong demand (30% likely) giving a profit of ₹40,000, normal demand (50%) giving ₹15,000, and weak demand (20%) leaving a loss of ₹12,000 after clearance.
Strong: 0.30 × 40,000
₹12,000
Normal: 0.50 × 15,000
₹7,500
Weak: 0.20 × (−12,000)
−₹2,400
Probabilities sum
0.30 + 0.50 + 0.20 = 1
Expected profit
₹17,100
12,000 + 7,500 − 2,400 = 17,100
The expected figure of ₹17,100 is not any of the three outcomes. In a single season she will earn ₹40,000, ₹15,000 or lose ₹12,000. The ₹17,100 is what repeating similar seasons would average out to, and it is the number to compare against not stocking, which has an expected value of zero.
Variation in outcomes still matters. Two options with the same expected value can differ greatly in how bad the bad case is, and a shop with thin cash may reasonably avoid the one that could lose ₹12,000.
Another angle: a raffle ticket
Expected value also shows why some games cost more than they return. A club raffle sells 1,000 tickets at ₹50. The prizes are one of ₹10,000, five of ₹1,000 and twenty of ₹250.
Total prize money
10,000 + 5 × 1,000 + 20 × 250 = ₹20,000
Average prize per ticket
20,000 ÷ 1,000 = ₹20
Ticket price
₹50
Expected net per ticket
−₹30
The raffle is a fundraiser; on average each ticket loses ₹30.
A negative expected value does not make a ticket 'wrong' to buy when the aim is to support the cause. It simply states the maths clearly, which is the main job of the calculation.
Comparing a safe option with a risky one
Expected value makes two different kinds of choice comparable. Option A pays ₹10,000 for certain. Option B pays ₹20,000 with probability 0.6 and nothing otherwise.
Option A
₹10,000 × 1 = ₹10,000
Option B
₹20,000 × 0.6 + ₹0 × 0.4 = ₹12,000
Difference
₹2,000 in B's favour
Higher expected value
Option B
But B gives ₹0 in four attempts out of ten
Whether B is better depends on whether you can live with the zero. Someone who needs at least ₹10,000 next month for a bill would sensibly take A; a business doing many such deals can take B. The expected value is the same for both, and the appetite for risk is what separates them.
Where expected value misleads
- One-off decisions. If you can only play once, the average is not what you will get, and the spread of results matters as much.
- Large, unaffordable losses. A 1% chance of losing everything can have a positive expected value yet still be a bad idea if you cannot absorb it.
- Poorly estimated probabilities. The answer is only as good as the chances you feed in; made-up percentages give made-up precision.
- Money that does not feel linear. The tenth lakh rupees of profit is worth less to most people than the first, which is why insurance has a positive price even though its expected value, for the buyer, is negative.
For these reasons, experts often look at expected value alongside the best case, the worst case and the standard deviation of outcomes. It is a starting point for the decision, not the decision itself.
The break-even probability
When a gamble has a cost to enter and a prize if it works, you can work out the probability at which it breaks even. This is useful for judging whether a quoted chance is worth paying for.
- cost:
- what you pay to take the chance
- prize:
- what you receive on success
Cost to enter a pitch competition
₹1,000
Prize
₹5,000
Break-even chance
1,000 ÷ 5,000
Break-even probability
20%
If you believe your chance is above 20%, the entry has a positive expected value
The same formula answers how big a prize must be: with a 5% chance of winning, the prize needs to exceed ₹1,000 ÷ 0.05 = ₹20,000 before entry is worthwhile in expectation. It also shows how sensitive the answer is to an estimate of the chance, which is rarely exact.
Quick uses, and where the probabilities come from
| Situation | Outcomes and weights | What EV tells you |
|---|---|---|
| Insurance premium vs claim | Chance of claim × claim size | Price is above the expected claim, because the insurer covers costs |
| Extended warranty | Chance of failure × repair cost | Whether the fee beats self-insuring |
| Job offers | Base pay + chance of bonus × bonus | Comparable annual value |
| Marketing test | Conversion rate × margin per sale | Spend you can justify per click |
In every case the process is the same: list outcomes, attach probabilities, multiply, add. A calculator is handy when there are five or more outcomes, or when the numbers are awkward decimals.
The hardest input is not the arithmetic but the probabilities. They can come from past frequencies (three strong seasons out of ten), from a model (a fair die gives 1 in 6), or from an informed guess. Past frequencies are better than gut feel when there are enough observations, and a guess is better than ignoring the question altogether, provided you test how much the answer changes if the guess is wrong.
Common questions
What is the formula for expected value?
Expected value is the sum of each outcome multiplied by its probability: E(X) = Σ x × p. For outcomes of ₹40,000, ₹15,000 and −₹12,000 with probabilities 0.3, 0.5 and 0.2, it is ₹17,100.
Does expected value tell me what will happen?
No. It is the long-run average over many repetitions. In a single trial you will see one of the actual outcomes, which may be far from the average. The spread of outcomes matters for one-off decisions.
Can expected value be negative?
Yes. It is negative when the probability-weighted losses exceed the gains. A ₹50 raffle ticket whose average prize is ₹20 has an expected value of −₹30. Negative values are common in games and insurance.
What should the probabilities add up to?
They must add up to 1, or 100%. If the total is less, you are missing an outcome; if more, a probability is too high. Check this before multiplying, since an error here ruins the result.
How is expected value different from the mean?
They are the same idea. The mean describes observed data; expected value applies the same weighted average to a probability distribution before the outcomes have occurred. Equal probabilities give the ordinary average.
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