Events, Unions, and Complements
Aligned to S-CP.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
The sample space is the set of all possible outcomes, and an event is a subset of it. Combine events using union (A or B), intersection (A and B), and complement (not A).
Worked examples
Example 1
Roll a die, so the sample space is {1, 2, 3, 4, 5, 6}. Let A = even = {2, 4, 6} and B = greater than 3 = {4, 5, 6}. Then A and B = {4, 6}, A or B = {2, 4, 5, 6}, and the complement of A is {1, 3, 5}.
Example 2
Draw one card from a standard deck. Let A = the card is red (26 outcomes) and B = the card is a face card (12 outcomes). Then A and B = red face cards = 6 outcomes. A or B = 26 + 12 − 6 = 32 outcomes. The complement of A is the 26 black cards.
Example 3
Common mistake: reading the complement of an event too narrowly. The complement of at least one head in three flips is NOT at least one tail — it is NO heads at all, the single outcome TTT. So P(at least one head) = 1 − 1/8 = 7/8. Whenever a problem says at least one, computing the complement is usually far quicker than listing every qualifying outcome.
Practice questions
- List the sample space for flipping two coins.
- For a die roll, list the event "less than 3."
- If A = {1, 2, 3} on a die, list the complement of A.
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
Eventos, uniones y complementos
El espacio muestral es el conjunto de todos los resultados posibles, y un evento es un subconjunto de él. Combina eventos usando la unión (A o B), la intersección (A y B) y el complemento (no A).
Ejemplo 1: Lanza un dado, así que el espacio muestral es {1, 2, 3, 4, 5, 6}. Sea A = par = {2, 4, 6} y B = mayor que 3 = {4, 5, 6}. Entonces A y B = {4, 6}, A o B = {2, 4, 5, 6}, y el complemento de A es {1, 3, 5}.
Ejemplo 2: Saca una carta de una baraja estándar. Sea A = la carta es roja (26 resultados) y B = la carta es una figura (12 resultados). Entonces A y B = figuras rojas = 6 resultados. A o B = 26 + 12 − 6 = 32 resultados. El complemento de A son las 26 cartas negras.
Ejemplo 3: Error común: leer el complemento de un evento de forma demasiado estrecha. El complemento de al menos una cara en tres lanzamientos NO es al menos una cruz — es NINGUNA cara, el único resultado XXX. Así que P(al menos una cara) = 1 − 1/8 = 7/8. Siempre que un problema diga al menos uno, calcular el complemento suele ser mucho más rápido que listar todos los resultados que cumplen.
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