HIFreeTools
ENES

Expected Value Calculator

Expected value (EV) is the probability-weighted average of all possible outcomes, showing what you can expect on average over many trials. Use this calculator to add any number of outcome–probability pairs and instantly compute the EV.

Enter each possible outcome and its probability (0 to 1). Probabilities must sum to 1.

Outcome ValueProbability (0–1)
Enter at least one valid outcome and probability to see the expected value.

Formula

EV = x₁p₁ + x₂p₂ + … + xₙpₙ

Each outcome value (xᵢ) is multiplied by its probability (pᵢ), and all products are summed. Probabilities must total 1 for the result to be valid.

Quick answer

Expected value is calculated by multiplying each possible outcome by its probability, then summing all those products: EV = Σ (xᵢ × pᵢ). All probabilities must sum to 1 (or 100%). A positive EV means the event is profitable on average; a negative EV means a net loss over time. Expected value is widely used in probability theory, gambling analysis, finance, insurance pricing, and decision-making under uncertainty.

Formula & method

EV = Σ (xᵢ × pᵢ) = x₁p₁ + x₂p₂ + … + xₙpₙ
  • EV — Expected value (weighted average outcome)
  • xᵢ — Value of the i-th outcome
  • pᵢ — Probability of the i-th outcome (0 to 1)
  • n — Total number of possible outcomes

Sum of each outcome multiplied by its probability. All probabilities must sum to 1.

Examples

Example 1: Fair Coin Flip Game
Input
Heads: win export const batch4Tools: ToolContent[] = 0 (probability 0.5); Tails: lose $5 (probability 0.5)
Result
EV = $2.50
Why
EV = (10 × 0.5) + (−5 × 0.5) = 5.00 − 2.50 = $2.50. Because the expected value is positive, this game is profitable on average over many flips.
Example 2: Dice Roll with Three Outcomes
Input
Roll 1–2: lose $3 (prob 0.3333); Roll 3–4: win $2 (prob 0.3333); Roll 5–6: win $8 (prob 0.3333)
Result
EV ≈ $2.33
Why
EV = (−3 × 0.3333) + (2 × 0.3333) + (8 × 0.3333) = −1.00 + 0.67 + 2.67 = $2.33. The expected gain per roll is about $2.33.
Example 3: Insurance Policy Payout
Input
No claim: $0 payout (prob 0.95); Minor claim: export const batch4Tools: ToolContent[] = ,500 payout (prob 0.04); Major claim: export const batch4Tools: ToolContent[] = 5,000 payout (prob 0.01)
Result
EV = $210.00
Why
EV = (0 × 0.95) + (1500 × 0.04) + (15000 × 0.01) = 0 + 60 + 150 = $210. If the annual premium is $500, the insurer earns an average of $290 per policy per year.
Example 4: Lottery Ticket Analysis
Input
Jackpot export const batch4Tools: ToolContent[] = ,000,000 (prob 0.000001); export const batch4Tools: ToolContent[] = 00 prize (prob 0.001); export const batch4Tools: ToolContent[] = 0 prize (prob 0.01); No win: $0 (prob 0.988999). Ticket cost: $2.
Result
Net EV = −$0.80
Why
Gross EV = (1000000 × 0.000001) + (100 × 0.001) + (10 × 0.01) + (0 × 0.988999) = 1.00 + 0.10 + 0.10 + 0 = export const batch4Tools: ToolContent[] = .20. Subtracting the $2 ticket cost gives a net EV of −$0.80 per ticket, a typical negative-EV game.

Frequently asked questions

What does expected value mean?

Expected value is the long-run average outcome of a random event. It tells you what value you can 'expect' on average if you repeated the same experiment many times. A positive EV favors the player; a negative EV favors the house or represents a net cost.

Do my probabilities need to add up to 1?

Yes. In a valid probability distribution, all outcome probabilities must sum to exactly 1 (or 100% if you enter percentages). If they don't sum to 1, your model is incomplete and the expected value will be inaccurate. This calculator warns you when the total deviates from 1.

Can expected value be negative?

Absolutely. A negative expected value means the average outcome is a loss. Most casino games, lottery tickets, and some insurance products have negative EV for the buyer, which is why they are profitable for the seller over thousands of transactions.

How many outcomes can I enter?

You can enter as many outcome–probability pairs as needed. The calculator starts with two rows and lets you add more with the 'Add Outcome' button. Each row requires a numeric value and a probability between 0 and 1.

What is expected value used for in real life?

Expected value is used in investing and portfolio analysis, insurance premium pricing, game theory, clinical trial design, engineering risk assessment, poker strategy, and business decision analysis. Any time you face a choice with uncertain outcomes, EV helps quantify the best option on average.

Is expected value the same as average or mean?

Expected value is a type of weighted mean where each possible value is weighted by its probability. For a fair (uniform) distribution where all outcomes are equally likely, the expected value equals the simple arithmetic mean. In general, EV accounts for outcomes that may occur with different frequencies.

Sources & references

External references open in a new tab. We are independent and not affiliated with these organizations.

  • ✓ Free to use
  • ✓ No sign-up required
  • ✓ Runs entirely in your browser — nothing is uploaded.
  • ✓ Formula and method shown above
Last updated: How we calculateReport a correctionDisclaimer

Provided “as is” for general information only — results may be inaccurate, so verify before you rely on them. No warranty; use at your own risk.

Built and reviewed by HIFreeTools against the formula shown above and any authoritative references cited on this page. See our methodology and editorial standards.

Related tools

Embed this tool on your site

Free to embed, no sign-up. Paste this code where you want the expected value calculator to appear: