For a theoretical probability you need to count all the outcomes. With two or three steps, like flipping two coins, that takes an organized list.
Idea clave
The sample space is the list of all possible outcomes. Organize it with a list, a table, or a tree diagram. When each step has a fixed number of choices, multiply: a coin and a number cube give $2 \times 6 = 12$ outcomes.
Por qué funciona
A tree branches at every step. Each of the 2 coin branches splits into 6 number-cube branches, so there are $2 \times 6 = 12$ paths. Each path is one outcome. Order matters: "heads on the first coin, tails on the second" is a different path from "tails, then heads."
Worked examples
Ejemplo resuelto: Two coins
Two coins are flipped. How many outcomes have exactly one head?
- Sample space: HH, HT, TH, TT.
- Exactly one head: HT and TH. These are different outcomes (which coin shows the head).
Respuesta: 2 outcomes
Ejemplo resuelto: Outfits
Ana has 3 shirts and 4 pairs of pants. How many outfits of one shirt and one pair of pants can she make?
- Each shirt can go with each of the 4 pairs of pants.
- $3 \times 4 = 12$ outfits.
Respuesta: 12
Ejemplo resuelto: Two number cubes
Two number cubes are rolled. How many outcomes have a sum of 7?
- The sample space is a 6-by-6 table: 36 outcomes.
- Sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1).
Respuesta: 6 outcomes
Error común
Listing "one head and one tail" only once. HT and TH are two different outcomes, which is why one head and one tail ($\frac{2}{4} = \frac{1}{2}$) is twice as likely as two heads ($\frac{1}{4}$). Label the coins (first and second) to keep them apart.