Conditional Probability
Aligned to S-CP.A.3 — Common Core State Standards for Mathematics.
What this lesson teaches
The conditional probability P(A given B) = P(A and B) / P(B) is the chance of A once B is known to have occurred. If P(A given B) = P(A), then A and B are independent.
Worked examples
Example 1
In a class, P(plays a sport and plays music) = 0.2 and P(plays music) = 0.5. Then P(plays a sport given plays music) = 0.2 / 0.5 = 0.4.
Example 2
Of 200 students, 120 take Spanish, 80 take art, and 50 take both. Find P(takes art | takes Spanish). Restrict to the 120 Spanish students; of those, 50 also take art. So the probability is 50/120 which is about 0.42. Notice the denominator shrank from 200 to 120 — conditioning narrows the world you are counting in.
Example 3
Common mistake: leaving the original total in the denominator. Computing 50/200 = 0.25 answers P(takes both), not P(art | Spanish). The condition after the bar REPLACES the sample space: once you know the student takes Spanish, only those 120 students remain possible. Circle the conditioning group first, then count inside it.
Practice questions
- Write the formula for P(A given B).
- If P(A and B) = 0.15 and P(B) = 0.3, find P(A given B).
- If P(A given B) = P(A), what does that tell you about the two events?
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
Probabilidad condicional
La probabilidad condicional P(A dado B) = P(A y B) / P(B) es la probabilidad de A una vez que se sabe que B ocurrió. Si P(A dado B) = P(A), entonces A y B son independientes.
Ejemplo 1: En una clase, P(practica un deporte y toca música) = 0.2 y P(toca música) = 0.5. Entonces P(practica un deporte dado que toca música) = 0.2 / 0.5 = 0.4.
Ejemplo 2: De 200 estudiantes, 120 toman español, 80 toman arte y 50 toman ambos. Halla P(toma arte | toma español). Restríngete a los 120 estudiantes de español; de ellos, 50 también toman arte. Así que la probabilidad es 50/120, aproximadamente 0.42. Observa que el denominador se redujo de 200 a 120 — condicionar estrecha el mundo en el que cuentas.
Ejemplo 3: Error común: dejar el total original en el denominador. Calcular 50/200 = 0.25 responde a P(toma ambos), no a P(arte | español). La condición después de la barra REEMPLAZA el espacio muestral: una vez que sabes que el estudiante toma español, solo esos 120 siguen siendo posibles. Encierra primero el grupo condicionante y luego cuenta dentro de él.
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