Expected Value
Aligned to S-MD.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
This is a (+) advanced standard. The expected value of a random variable is the sum of each value times its probability: E(X) = Σ x × P(x). It is the long-run average, or mean, of the distribution.
Worked examples
Example 1
A game pays $1, $2, or $5 with probabilities 0.5, 0.3, and 0.2. Then E(X) = 1 × 0.5 + 2 × 0.3 + 5 × 0.2 = 0.5 + 0.6 + 1.0 = $2.10 per play on average.
Example 2
An extended warranty costs $80. The company estimates a 5% chance of paying a $600 claim and a 2% chance of paying a $1,500 claim. Expected payout = 0.05(600) + 0.02(1500) = 30 + 30 = $60. Charging $80 leaves an expected profit of $20 per warranty — which is how the price was set.
Example 3
Common mistake: expecting the expected value to be a possible outcome. A game paying $1, $2 or $5 has E(X) = $2.10, but no single play ever pays $2.10. Expected value is the long-run AVERAGE over many trials, not a prediction of one trial. And it says nothing about risk: a coin flip paying $0 or $100 and a flat payment of $50 have the same expected value and are nothing alike.
Practice questions
- Write the formula for the expected value of a random variable.
- A die roll pays its face value in dollars. Find E(X).
- A raffle ticket wins $100 with probability 0.01 and $0 otherwise. Find its expected value.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Valor esperado
Este es un estándar avanzado (+). El valor esperado de una variable aleatoria es la suma de cada valor por su probabilidad: E(X) = Σ x × P(x). Es el promedio a largo plazo, o la media, de la distribución.
Ejemplo 1: Un juego paga $1, $2 o $5 con probabilidades 0.5, 0.3 y 0.2. Entonces E(X) = 1 × 0.5 + 2 × 0.3 + 5 × 0.2 = 0.5 + 0.6 + 1.0 = $2.10 por jugada en promedio.
Ejemplo 2: Una garantía extendida cuesta $80. La compañía estima un 5% de probabilidad de pagar un reclamo de $600 y un 2% de pagar uno de $1,500. Pago esperado = 0.05(600) + 0.02(1500) = 30 + 30 = $60. Cobrar $80 deja una ganancia esperada de $20 por garantía — así se fijó el precio.
Ejemplo 3: Error común: esperar que el valor esperado sea un resultado posible. Un juego que paga $1, $2 o $5 tiene E(X) = $2.10, pero ninguna jugada paga jamás $2.10. El valor esperado es el PROMEDIO a largo plazo en muchos intentos, no una predicción de un intento. Y no dice nada del riesgo: un volado que paga $0 o $100 y un pago fijo de $50 tienen el mismo valor esperado y no se parecen en nada.
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