What is the expected value formula?

Expected value is E[X] = Σ x·p — multiply each possible outcome x by its probability p, then add up all those products. For a fair six-sided die each face has probability 1/6, so E[X] = 1·(1/6) + 2·(1/6) + … + 6·(1/6) = 21/6 = 3.5. The expected value is the long-run average you would see if you repeated the experiment many times, even when no single trial can land exactly on 3.5.

Do my probabilities have to add up to exactly 1?

For a true probability distribution, yes — the probabilities must sum to 1. This calculator checks that sum and flags it when it is off. If your numbers do not add to 1 it still gives an answer by treating the column as weights and computing the weighted mean Σ x·p ÷ Σp, which is handy when you enter counts like 50 and 50 instead of 0.5 and 0.5. The normalize button rescales everything so the sum becomes 1 and you can read true probabilities.

How do you calculate the variance and standard deviation?

Variance measures spread: Var = Σ p·(x − E)², where E is the expected value. For a fair die E = 3.5 and Var = 35/12 ≈ 2.917. The standard deviation is just the square root of the variance, about 1.708 for the die, and it is in the same units as the outcomes, which makes it easier to interpret than the variance. A larger standard deviation means the outcomes are more spread out around the mean.

Why is a casino bet a negative expected value?

Take $1 on red in American roulette: you win $1 with probability 18/38 and lose $1 with probability 20/38, so E[X] = 1·(18/38) + (−1)·(20/38) = −2/38 ≈ −$0.053 per spin. The expected value is negative because the two green zeros tilt the odds toward the house. Over thousands of spins that small negative edge is exactly how the casino profits, even though any single spin can win.

How is expected value used to make decisions?

Decision-makers compare the expected value of each option and, all else equal, pick the highest. Suppose project A returns $50,000 with probability 0.25 and $0 otherwise (E = $12,500) while project B returns a sure $10,000 (E = $10,000). On expected value A wins, but the variance also matters: A is far riskier. Pairing E[X] with the standard deviation from this tool lets you weigh average payoff against the spread before committing.

What is the difference between expected value and a plain average?

A plain average weights every data point equally: add the numbers and divide by the count. Expected value weights each outcome by its probability, so a rare big payoff counts less than a likely small one. If all probabilities are equal the two coincide. The simple average of 1 through 6 is also 3.5, matching the fair die — but a loaded die with unequal probabilities has the same possible values and a different expected value.

Check a probability homework answer

A stats assignment asks for the expected value of a discrete random variable: outcomes 0, 1, 2, 3 with probabilities 0.1, 0.3, 0.4, 0.2. Type them in, read E[X] = 1.7, and confirm the variance and standard deviation in the same view. Because probabilities accept fractions, the fair-die warm-up problem (E = 3.5) takes seconds, and you can paste the whole list as text instead of clicking row by row.

Decide whether a bet is worth it

Before putting money down, model the bet as outcomes and probabilities. A $1 stake on red in American roulette pays +1 with probability 18/38 and −1 with probability 20/38, and the tool shows E[X] ≈ −$0.053. Seeing a negative expected value in black and white is the clearest way to understand why the house always wins over the long run, no matter how a single spin goes.

Compare two risky business decisions

Project A might earn $400,000 (probability 0.05), $50,000 (probability 0.25) or nothing. Project B earns a steady $15,000. Enter each project as its own set of outcomes, compare the expected values, and look at the standard deviation to judge risk. The higher expected value is not always the right call once you weigh how much the outcomes swing around that average.

Price an insurance policy or warranty

An extended warranty pays out $4,800 with probability 0.03 and collects a $200 premium otherwise. Model the insurer's net as −200 with probability 0.97 and +4,800 with probability 0.03 to see the expected value per policy. That single number tells you whether the premium covers the expected claim cost with room for profit and administration.

Expected Value Calculator — E[X], Variance and Standard Deviation

E[X] = Σ x·p — enter outcomes and probabilities, get expected value, variance and standard deviation, all in your browser

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Outcomes and their probabilities

Value x	Probability p

Try an example

Or paste as text (one pair per line: value : probability)

Separators :, comma or space all work. Probabilities can be fractions like 1/6.

Probabilities sum to 1 — a valid distribution.

Results

Expected value E[X]

3.5

E[X] = Σ x·p

Probability sum Σp

Variance Var = Σ p·(x − E)²

2.91667

Standard deviation σ

1.70783

Positive expected value (3.5): on average this bet gains money each play.

Every value and probability stays in your browser tab. Nothing is uploaded, logged, or stored.

What this tool does

Free expected value calculator for discrete random variables. Type each possible outcome with its probability and the tool returns E[X] = Σ x·p, the variance Var = Σ p·(x − E)² and the standard deviation, instantly and entirely in your browser. Probabilities accept fractions like 1/6, so the classic fair-die case (E = 3.5) takes three seconds to verify. If your probabilities do not add to 1 the tool still works: it treats the column as weights and computes a weighted mean, and a one-click button normalizes everything to a clean distribution. Built for students checking homework, gamblers sizing up a bet, analysts comparing risky decisions and insurers pricing a claim. Paste a list as text or fill the table row by row, copy the result, and share a link that reopens your exact inputs. 100% client-side, nothing is uploaded or logged.

Tool details

Input: Text + Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 9 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Expected Value Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Check a probability homework answer
A stats assignment asks for the expected value of a discrete random variable: outcomes 0, 1, 2, 3 with probabilities 0.1, 0.3, 0.4, 0.2. Type them in, read E[X] = 1.7, and confirm the variance and standard deviation in the same view. Because probabilities accept fractions, the fair-die warm-up problem (E = 3.5) takes seconds, and you can paste the whole list as text instead of clicking row by row.
Decide whether a bet is worth it
Before putting money down, model the bet as outcomes and probabilities. A $1 stake on red in American roulette pays +1 with probability 18/38 and −1 with probability 20/38, and the tool shows E[X] ≈ −$0.053. Seeing a negative expected value in black and white is the clearest way to understand why the house always wins over the long run, no matter how a single spin goes.
Compare two risky business decisions
Project A might earn $400,000 (probability 0.05), $50,000 (probability 0.25) or nothing. Project B earns a steady $15,000. Enter each project as its own set of outcomes, compare the expected values, and look at the standard deviation to judge risk. The higher expected value is not always the right call once you weigh how much the outcomes swing around that average.
Price an insurance policy or warranty
An extended warranty pays out $4,800 with probability 0.03 and collects a $200 premium otherwise. Model the insurer's net as −200 with probability 0.97 and +4,800 with probability 0.03 to see the expected value per policy. That single number tells you whether the premium covers the expected claim cost with room for profit and administration.

Common pitfalls

Forgetting that probabilities must sum to 1. If they add to 0.9 you are missing an outcome or mistyped a value, and the raw E[X] is off. The tool flags the sum and offers a normalize button — but check whether a missing case is the real reason before normalizing away the gap.
Treating expected value as a value you will actually get. E[X] for a fair die is 3.5, which never appears on a single roll. Expected value is a long-run average across many trials, not a prediction for one trial — useful for pricing and decisions, not for guessing the next outcome.
Confusing variance and standard deviation. Variance is in squared units; standard deviation is its square root and shares the units of the outcomes. Reporting a variance of 25 when you meant a spread of 5 (the standard deviation) overstates the risk by a factor you did not intend.

Privacy

Every calculation — the expected value, the variance, the standard deviation and the probability sum — is plain JavaScript that runs in your browser tab. Your outcomes and probabilities never leave the page and nothing is logged. The one caveat: the shareable URL encodes your table in the query string, so a share link pasted into chat will record those numbers in the recipient server's access log. If the values are sensitive, use the copy button and paste the text rather than sharing the link.

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Expected Value Calculator — E[X], Variance and Standard Deviation

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Expected Value Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Check a probability homework answer

Decide whether a bet is worth it

Compare two risky business decisions

Price an insurance policy or warranty

Common pitfalls

Privacy

FAQ

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