Random Variables and Probability Distributions
Aligned to S-MD.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
This is a (+) advanced standard. A random variable assigns a number to each outcome of a chance process. Its probability distribution lists each value with its probability; all probabilities are between 0 and 1 and together sum to 1.
Worked examples
Example 1
Let X be the number of heads in two coin flips. The outcomes give TT → 0, HT and TH → 1, and HH → 2, so the distribution is P(0) = 1/4, P(1) = 1/2, P(2) = 1/4. These probabilities sum to 1.
Example 2
Let X be the number rolled on a fair six-sided die. The distribution is P(X = k) = 1/6 for each k from 1 to 6, and the six probabilities sum to 1. Now let Y be the LARGER of two dice: P(Y = 1) = 1/36, P(Y = 2) = 3/36, P(Y = 3) = 5/36, and so on up to P(Y = 6) = 11/36 — the same sample space, a very different distribution.
Example 3
Common mistake: assigning probabilities that do not sum to 1. If a spinner is claimed to land on red 0.5, blue 0.3 and green 0.4, the total is 1.2 and the model is impossible. Every probability distribution must have all values between 0 and 1 AND a total of exactly 1. Adding the column is the fastest check that a distribution is legitimate before you compute anything with it.
Practice questions
- Define a random variable for the roll of one die.
- For two coin flips, what is P(X = 1) where X is the number of heads?
- The probabilities in any distribution must sum to what 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
Variables aleatorias y distribuciones de probabilidad
Este es un estándar avanzado (+). Una variable aleatoria asigna un número a cada resultado de un proceso al azar. Su distribución de probabilidad enumera cada valor con su probabilidad; todas las probabilidades están entre 0 y 1 y juntas suman 1.
Ejemplo 1: Sea X el número de caras en dos lanzamientos de moneda. Los resultados dan TT → 0, CX y XC → 1, y CC → 2, así que la distribución es P(0) = 1/4, P(1) = 1/2, P(2) = 1/4. Estas probabilidades suman 1.
Ejemplo 2: Sea X el número obtenido al lanzar un dado justo de seis caras. La distribución es P(X = k) = 1/6 para cada k de 1 a 6, y las seis probabilidades suman 1. Ahora sea Y el MAYOR de dos dados: P(Y = 1) = 1/36, P(Y = 2) = 3/36, P(Y = 3) = 5/36, y así hasta P(Y = 6) = 11/36 — el mismo espacio muestral, una distribución muy distinta.
Ejemplo 3: Error común: asignar probabilidades que no suman 1. Si se afirma que una ruleta cae en rojo con 0.5, azul con 0.3 y verde con 0.4, el total es 1.2 y el modelo es imposible. Toda distribución de probabilidad debe tener todos sus valores entre 0 y 1 Y un total de exactamente 1. Sumar la columna es la comprobación más rápida de que una distribución es legítima antes de calcular nada con ella.
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