# Expected value calculator

> Calculate expected value E[X], variance and standard deviation for a bet, game or decision from its outcomes and their probabilities or weights.

ఇంటరాక్టివ్ వెర్షన్: https://www.calcopenly.com/te/statistics/expected-value-calculator
విషయం: సంఖ్యాశాస్త్రం మరియు సంభావ్యత కాలిక్యులేటర్లు

Expected value is the probability-weighted average of the outcomes, E[X] = Σ xᵢpᵢ: what a bet, game or decision returns per play over many plays. The variance, Σ(xᵢ − E[X])²pᵢ, and its square root, the standard deviation, measure how far single results swing around that average. Counts or relative weights are rescaled to probabilities before the sums.

The default is a $5 scratch card that loses the stake with probability 0.8 and wins a net $5, $20 or $95 with probabilities 0.15, 0.04 and 0.01. Its expected value is −$1.50 per card, even though 20% of cards come out ahead.

Enter each outcome as a net result, winnings minus the stake: a lost $5 stake is −5 and a $100 prize on a $5 card is 95. The probabilities must add up to 1, or switch to counts and weights.

## ఇన్‌పుట్‌లు

- **Outcomes (values or payoffs)**: Net result of each outcome, e.g. −5 for losing a $5 stake.
- **Probabilities**: One per outcome, in the same order. Fractions such as 1/38 work.
- **Second list holds** (ఎంపికలు: Probabilities (sum to 1), Counts or weights)

## ఫలితాలు

- Expected value E[X] — ప్రధాన ఫలితం
- Variance
- Standard deviation
- Probability of a positive outcome
- Worst outcome
- Best outcome

## సూత్రం

$$
E[X] = \sum_i x_i\,p_i,\qquad \operatorname{Var}(X) = \sum_i (x_i - E[X])^2\,p_i
$$

## పరిష్కరించిన ఉదాహరణలు

### $5 scratch card (defaults)

- Outcomes (values or payoffs): -5, 5, 20, 95
- Probabilities: 0.8, 0.15, 0.04, 0.01
- Second list holds: Probabilities (sum to 1)
- **Expected value E[X]: -1.5**
- **Variance: 127.75**
- **Probability of a positive outcome: 0.2**
- తనిఖీ చేసిన మూలం: Hand calculation: −4 + 0.75 + 0.8 + 0.95 = −1.5; Σp(x − μ)² with Python fractions = 127.75

### American roulette, single-number bet

- Outcomes (values or payoffs): 35, -1
- Probabilities: 1/38, 37/38
- Second list holds: Probabilities (sum to 1)
- **Expected value E[X]: -0.052632**
- తనిఖీ చేసిన మూలం: House edge of a straight-up bet: −2/38 = −5.26% (standard roulette tables)

### Fair die

- Outcomes (values or payoffs): 1 2 3 4 5 6
- Probabilities: 1 1 1 1 1 1
- Second list holds: Counts or weights
- **Expected value E[X]: 3.5**
- **Variance: 2.916667**
- తనిఖీ చేసిన మూలం: E = 7/2 and Var = 35/12 for a fair die (textbook result; Python fractions)

### Edge case: a certain outcome

- Outcomes (values or payoffs): 42
- Probabilities: 1
- Second list holds: Probabilities (sum to 1)
- **Expected value E[X]: 42**
- **Variance: 0**
- **Standard deviation: 0**
- తనిఖీ చేసిన మూలం: A single outcome with probability 1 has no spread

## ప్రశ్నలు

### How do you calculate expected value?

Multiply each outcome by its probability and add the products: E[X] = Σ xᵢpᵢ. For a fair six-sided die, E = (1 + 2 + 3 + 4 + 5 + 6) × 1/6 = 3.5. For the default scratch card, −5 × 0.8 + 5 × 0.15 + 20 × 0.04 + 95 × 0.01 = −1.5, a loss of $1.50 per $5 card on average.

### What does a negative expected value mean?

The bet loses money on average per play. The −$1.50 scratch card returns −30% of its $5 price, so 100 cards are expected to lose $150. Single results vary: with a standard deviation of 11.30 per card, the total over 100 independent cards has a standard deviation of 11.30 × √100 = 113, so some buyers of 100 cards still come out ahead.

### What is the house edge in roulette?

5.26% on an American double-zero wheel and 2.70% on a European single-zero wheel. A single-number bet pays 35 to 1. With 38 pockets, the expected value per unit staked is (35 − 37)/38 = −2/38 = −0.0526; with 37 pockets it is (35 − 36)/37 = −1/37 = −0.0270. Almost every other American bet has the same 5.26% edge.

### Is expected value the most likely outcome?

No. It is a long-run average and may not be a possible result at all: a die's expected value is 3.5, which no roll shows. The scratch card's expected value is −1.5, but its most likely outcome is losing the full $5, which happens 80% of the time. In a skewed payoff like this, a few large prizes pull the average above the typical result.

### Is the option with the highest expected value always the best choice?

Not always, because expected value ignores risk. A sure $50 and a 50% chance of $100 both have an expected value of 50, but standard deviations of 0 and 50. Insurance has a negative expected value for the buyer yet is rational when it removes a loss the buyer could not absorb. Expected utility theory (von Neumann and Morgenstern, 1944) formalizes this trade-off.

### “Expected value calculator” ఎంత కచ్చితమైనది?

కచ్చితత్వం మీ ఇన్‌పుట్‌లు, పద్ధతిలోని ఊహలపై ఆధారపడి ఉంటుంది. దశాంశ గణన 50 సార్థక అంకెలను ఉపయోగిస్తుంది. అంచనాలు, సంఖ్యా పద్ధతులు, మూల డేటా తక్కువ కచ్చితత్వంతో ఉండవచ్చు; ప్రదర్శనలో విలువలను రౌండ్ చేయడం ఈ పరిమితులను తొలగించదు. స్వతంత్ర మూలాల పరిష్కారాలతో తనిఖీ చేసిన ఉదాహరణలు: 4. ఉదాహరణకు, “$5 scratch card (defaults)”ను Hand calculation: −4 + 0.75 + 0.8 + 0.95 = −1.5; Σp(x − μ)² with Python fractions = 127.75తో తనిఖీ చేశారు.

### ఈ పద్ధతికి మూలం ఏమిటి?

Grinstead & Snell, Introduction to Probability (AMS), chapter 6 Expected value and variance; NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.1 What is a probability distribution.

## మూలాలు

- [Grinstead & Snell, Introduction to Probability (AMS), chapter 6 Expected value and variance](https://math.dartmouth.edu/~prob/prob/prob.pdf)
- [NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.1 What is a probability distribution](https://www.itl.nist.gov/div898/handbook/eda/section3/eda361.htm)