Calcylator
Probability

Expected value:
chance of win × win + chance of loss × loss

Weigh a two-outcome choice with the expected value formula, find the break-even chance, and see why one positive result does not guarantee a good decision.

Calcylator Editorial Team

Updated · 5 min read

The expected value formula for a win or a loss

For a choice with two outcomes, expected value equals the chance of winning times the win payoff, plus the chance of losing times the loss payoff. With a 30 percent chance of winning 500 and a 70 percent chance of losing 100, the expected value is 0.3 × 500 + 0.7 × (−100) = 80.

Expected value (two outcomes) =EV = p × win + (1 − p) × loss
p:
probability of winning, as a fraction (30% = 0.3)
win:
payoff if you win
loss:
payoff if you lose; enter it as a negative number for a loss
The Calcylator expected value calculator takes the win probability as a percentage from 0 to 100, then applies this formula.

  • Win probability

    30 %

  • Win payoff

    500

  • Loss payoff

    -100

  • Calculation

    0.30 × 500 + 0.70 × (−100)

Expected value

80

Expected value is a long-run average: you never actually get 80 in a single trial.

Finding the break-even probability

The break-even probability is the win chance at which the expected value is exactly zero. Rearranging the formula gives p = −loss ÷ (win − loss). Take a business bid that earns 300 if won and loses 150 on proposal costs if not. Break-even is 150 ÷ (300 + 150) = 33.3 percent.

Win 300, lose 150: expected value by win chance
Win chanceExpected value
20%-60
33.3%0
40%30
50%75
60%120

If you believe your real chance is above 33.3 percent the bid has a positive expected value, and if it is below, you are likely to lose money over many bids.

A ticket that looks cheap

A ₹100 ticket gives a 1 percent chance at a ₹5,000 prize. Your net result is a gain of ₹4,900 if you win and a loss of ₹100 if you do not. EV = 0.01 × 4,900 + 0.99 × (−100) = 49 − 99 = −50, so on average each ticket costs you ₹50.

Another way to see it: the prize is worth 0.01 × 5,000 = ₹50 in expectation, and you paid ₹100 for it. Always include the price of the ticket as part of the loss.

The same method helps a small business owner. A quote that wins 40 percent of the time with a profit of 300 and loses 150 in preparation costs otherwise has EV = 0.4 × 300 + 0.6 × (−150) = 30. It is positive, but thin, so a small rise in preparation cost could wipe it out.

What expected value does not tell you

  • It ignores risk. A positive EV can still ruin you if you can only afford one try.
  • It assumes the probability is right. A rough guess gives a rough answer.
  • It treats money as linear, but losing 10,000 can hurt more than winning 10,000 helps.
  • It describes the average over many repeats, not the next single outcome.

This is a general maths tool, not financial or investment advice. Real decisions have more than two outcomes and often cannot be repeated.

Common mistakes

  • Entering the loss as a positive number, which turns a cost into a gain.
  • Typing 0.3 into a field that expects a percentage; use 30.
  • Forgetting that the loss probability is 100 minus the win probability.
  • Leaving out the cost of playing when the payoff is only the prize.

Common questions

What is the expected value formula?

For two outcomes, EV = p × win + (1 − p) × loss, where p is the probability of winning. For a 30 percent chance of winning 500 and a 70 percent chance of losing 100, the expected value is 80.

How do I calculate expected value?

Multiply each outcome by its probability and add the results. With two outcomes, take the win chance times the win payoff, then add the loss chance times the loss payoff. Enter the loss as a negative number so it reduces the total.

What does a negative expected value mean?

It means that, on average, you lose money or value each time you repeat the choice. A ₹100 ticket with a 1 percent chance at ₹5,000 has an expected value of −₹50. Single outcomes can still be wins, but the long-run average is negative.

How do I find the break-even probability?

Use p = −loss ÷ (win − loss). If winning pays 300 and losing costs 150, break-even is 150 ÷ 450 = 33.3 percent. Above that win chance the expected value is positive; below it the expected value is negative.

Is expected value the same as the most likely result?

No. Expected value is a probability-weighted average, so it may be a figure that never actually occurs. In the 30 percent example above you either win 500 or lose 100, yet the expected value is 80, a result you can never get in one try.

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