Two-step experiments

Listing outcomes of two-step experiments with arrays and tree diagrams, with and without replacement.

Tables

Rolling two dice gives $6 \times 6 = 36$ equally likely outcomes. A table of sums shows that $7$ is the most likely total ($6$ ways).

Example 1●●

Two dice are rolled. Find $P(\text{sum} = 10)$.

Outcomes: $(4,6), (5,5), (6,4)$, so $3$ ways.

$P = \dfrac{3}{36} = \dfrac{1}{12}$

Each branch of a tree shows one stage. Multiply along the branches to get the probability of a path.

Example 2●●●

A bag has $3$ red and $2$ blue counters. Two are taken without replacement. Find $P(\text{both red})$.

$P(\text{red first}) = \dfrac{3}{5}$, then $P(\text{red second}) = \dfrac{2}{4}$.

$P = \dfrac{3}{5} \times \dfrac{2}{4} = \dfrac{6}{20} = \dfrac{3}{10}$

Flashcards

Outcomes for two dice

$36$

Outcomes for two coins

$4$: HH, HT, TH, TT

Along a tree path

Multiply the probabilities.

Practice

Two dice: find $P(\text{sum} = 10)$.
$3$ red, $2$ blue, two taken without replacement. Find $P(\text{both red})$.
Same bag, but with replacement. Find $P(\text{both red})$.
Two coins are tossed. Find $P(\text{two heads})$.
Two coins are tossed. Find $P(\text{exactly one head})$.
Two dice: find $P(\text{sum} = 7)$.
Two dice: find $P(\text{double})$.
$3$ red, $2$ blue, two taken without replacement. Find $P(\text{blue then red})$.