Expected Value Calculator: E(X), Variance & Expected Profit

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Formula at a glance

Expected Value Calculator: Expected value — E(X) = Σ x·P(x). Example: A simple prize game → Expected value E(X): 7.5 — How the Expected Value Calculator works: Expected value — E(X) = Σ x·P(x)

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📘 About the Expected Value Calculator

The expected value is the average outcome if you could repeat a random process many times: each outcome multiplied by its probability, then added. It decides whether a bet, lottery ticket, insurance policy or investment is worth it on average.

Enter the outcomes and their probabilities (or frequencies, which are converted). The calculator gives E(X), the variance and standard deviation, a table of x·p, and — with a cost to play — the expected profit or loss per play.

🧮 Formula

Expected value

E(X) = Σ x·P(x)

Variance

Var(X) = Σ x²·P(x) − [E(X)]²

✏️ Worked examples

A simple prize game

You enter
Values (outcomes)	0, 10, 100
Probabilities (or frequencies)	0.7, 0.25, 0.05
Cost to play (optional)	5

You get
Expected value E(X)	7.5
Variance	468.75
Standard deviation	21.65063509
E(X²)	525
Expected profit after paying 5	2.5 — on average you gain

E(X) = 7.5.

Fair die

You enter
Values (outcomes)	1, 2, 3, 4, 5, 6
Probabilities (or frequencies)	1, 1, 1, 1, 1, 1

You get
Expected value E(X)	3.5
Variance	2.916666667
Standard deviation	1.707825128
E(X²)	15.16666667

E(X) = 3.5.

🧭 How to use this calculator

What you enter

Values (outcomes)
Probabilities (or frequencies)
Cost to play (optional)for expected profit

What you get

Expected value E(X)
Variance
Standard deviation
E(X²)

❓ Frequently asked questions

What does expected value mean?

The average result per trial over the long run. A game with E = ₹2 profit per play wins ₹200 on average over 100 plays, though any single play may lose.

Is a lottery ticket worth buying?

Almost never on expected value: the expected prize is well below the ticket price.

Can I use frequencies instead of probabilities?

Yes — they are divided by their total automatically.

E[X] Expected Value Calculator (Mean of a Probability Distribution)