Events as Subsets of Outcomes
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. You combine events with unions (A or B), intersections (A and B), and complements (not A).
Worked examples
Example 1
Roll a fair die, sample space {1,2,3,4,5,6}. Let A = even = {2,4,6} and B = greater than 3 = {4,5,6}. Then A and B (intersection) = {4,6}, A or B (union) = {2,4,5,6}, and not A (complement) = {1,3,5}.
Example 2
Flip a coin three times. The sample space has 8 equally likely outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Let A = at least two heads = {HHH, HHT, HTH, THH}, so P(A) = 4/8 = 1/2. Let B = first flip is a tail = {THH, THT, TTH, TTT}. Then A and B = {THH}, so P(A and B) = 1/8.
Example 3
Common mistake: assuming every listed outcome is equally likely. Rolling two dice and listing the sums 2 through 12 gives 11 outcomes, but P(sum = 7) is 6/36 while P(sum = 2) is 1/36 — the sums are not equally likely because seven can happen six different ways. Build the sample space from the underlying equally likely events (the 36 ordered pairs), then group them.
Practice questions
- List the sample space for flipping two coins.
- For a die, write the event 'less than 3' as a subset.
- Given A = {1,2,3} and B = {2,3,4}, find A ∩ B and A ∪ B.
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 como Subconjuntos de Resultados
El espacio muestral es el conjunto de todos los resultados posibles, y un evento es un subconjunto de él. Combinas eventos con uniones (A o B), intersecciones (A y B) y complementos (no A).
Ejemplo 1: Lanza un dado justo, espacio muestral {1,2,3,4,5,6}. Sea A = par = {2,4,6} y B = mayor que 3 = {4,5,6}. Entonces A y B (intersección) = {4,6}, A o B (unión) = {2,4,5,6}, y no A (complemento) = {1,3,5}.
Ejemplo 2: Lanza una moneda tres veces. El espacio muestral tiene 8 resultados igualmente probables: CCC, CCX, CXC, CXX, XCC, XCX, XXC, XXX (C = cara, X = cruz). Sea A = al menos dos caras = {CCC, CCX, CXC, XCC}, así que P(A) = 4/8 = 1/2. Sea B = el primer lanzamiento es cruz = {XCC, XCX, XXC, XXX}. Entonces A y B = {XCC}, así que P(A y B) = 1/8.
Ejemplo 3: Error común: suponer que todos los resultados listados son igualmente probables. Lanzar dos dados y listar las sumas de 2 a 12 da 11 resultados, pero P(suma = 7) es 6/36 mientras que P(suma = 2) es 1/36 — las sumas no son igualmente probables porque el siete puede ocurrir de seis formas. Construye el espacio muestral con los eventos igualmente probables de base (los 36 pares ordenados) y luego agrúpalos.
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