♥ Support Post Sign in Join
Learn · Mathematics · Grade 11 · Probability

Conditional probability and independence

P(A | B), tree diagrams, the multiplication rule for dependent events, and testing whether two events are independent.

High school Lesson 31 of 31 Interactive figure Practice

First, the rules

Probability with extra information

"Given that" shrinks the world: once you know B happened, only the outcomes inside B count. Everything here is that one idea written three ways.

★ Conditional probability
Examples 1, 2
P(A | B) = P(A ∩ B)P(B)
Read "probability of A given B". The denominator is the new, smaller world.
★ Multiplication rule
Example 2
P(A ∩ B) = P(B) · P(A | B) = P(A) · P(B | A)
★ Counting version
Example 1
P(A | B) = |A ∩ B||B| — in a table, divide the cell by the row or column total you were "given".
★ Independent events
Example 3
P(A ∩ B) = P(A) · P(B) ⇔ P(A | B) = P(A)
Knowing B changes nothing about A. Test it with the product — do not guess from the story.
★ With and without replacement
Example 2
With replacement the draws are independent and the fractions repeat. Without replacement the second denominator drops by 1 — and so does the numerator if the first draw used up what you want.
Total probability
Example 3
P(A) = P(A | B) · P(B) + P(A | B′) · P(B′) — add up the branches of a tree diagram that end in A.

See it

Shrink the world

A hundred equally likely squares. Set how many are in A, in B and in both, then switch to "given B": everything outside B fades and the probability of A is counted inside B only.

Step by step

Worked examples

1

Reading a two-way table

counting version

In a class, 12 boys and 8 girls take the bus; 6 boys and 14 girls do not. A student is chosen at random. Find P(bus | girl) and P(girl | bus).

Ask yourself: The word after "given" tells you which row or column is your whole world.

Show the solution
  1. Girls: 8 + 14 = 22; of them 8 take the bus: P(bus | girl) = 822 = 411.
  2. Bus riders: 12 + 8 = 20; of them 8 are girls: P(girl | bus) = 820 = 25.
  3. Same cell (8), different denominators — the two conditionals are not the same number.

Answer 411 and 25

2

Two draws without replacement

multiplication rule

A bag holds 5 red and 3 blue marbles. Two are drawn without replacement. Find (a) P(both red), (b) P(second is red).

Ask yourself: (a) Multiply along one branch. (b) Add every branch that ends in red — or notice the symmetry.

Show the solution
  1. (a) P(R₁) · P(R₂ | R₁) = 58 · 47 = 2056 = 514.
  2. (b) Two branches end in red: 58 · 47 + 38 · 57 = 20 + 1556 = 3556 = 58.
  3. Exactly P(first is red): before you look at the first marble, the second is just as likely to be red.

Answer (a) 514 (b) 58

3

Testing independence

product test · total probability

Of all students 60% have a bike. Among bike owners 40% also take the bus; among the others 70% take the bus. (a) Find P(bus). (b) Are "has a bike" and "takes the bus" independent?

Ask yourself: (a) Tree diagram: two branches end in "bus". (b) Compare P(bus | bike) with P(bus).

Show the solution
  1. (a) P(bus) = 0.6 · 0.4 + 0.4 · 0.7 = 0.24 + 0.28 = 0.52.
  2. (b) P(bus | bike) = 0.4 ≠ 0.52 = P(bus) — knowing someone has a bike changes the chance, so the events are not independent. Equivalent test: P(bike ∩ bus) = 0.24 but P(bike) · P(bus) = 0.312.

Answer (a) 0.52 (b) not independent

Practice: conditional probability

0 / 0

Two-way tables, two draws without replacement, and the product test for independence. Your best score is kept in this browser only.

About this lesson

Conditional probability and independence: P(A | B) = P(A ∩ B) / P(B), reading it from a two-way table, tree diagrams for draws without replacement, the multiplication rule for dependent events, and the test for independence P(A ∩ B) = P(A) · P(B). Worked examples and practice with exact fractions.

Part of Learn mathematics · Grade 11 · Probability. Check a number with the scientific calculator or draw a function with the function plotter. Everything on this page runs in your browser; nothing you type is sent or stored, and your practice score stays in this browser only. Spotted an error? Tell us.