Tree Diagrams & Two-Way Tables (Grade 11)

Drawing a two-stage tree diagram, the difference between drawing with and without replacement, reading a two-way contingency table, and testing independence from one.

A tree diagram lays out every possible outcome of a two-stage experiment. Multiply along the branches; add between them.

1The two rules

Along a path (AND): multiply
Between paths (OR): add

Every set of branches from one point adds to 1

2Without replacement

A bag holds 5 red and 3 blue beads. Two are drawn without replacement — so the second draw has only 7 beads left, and the count of the colour taken has dropped by one.

Two beads drawn without replacement from 5 red and 3 blue. Note the denominators change from 8 to 7.

Worked ExampleFrom the tree diagram, calculate P(both red) and P(one of each)

1
P(RR) = 58 × 47 = 2056 = 514.
Multiply along the top path.
2
P(RB) = 58 × 37 = 1556.
3
P(BR) = 38 × 57 = 1556.
4
P(one of each) = 1556 + 1556 = 3056 = 1528.
Add the two paths that give one of each — 'or'.
5
Check: all four paths total 2056 + 1556 + 1556 + 656 = 1. ✓
Always add the leaves as a check.

⚠️

With replacement the denominator stays 8 and P(RR) = 58 × 58. Read the question carefully — 'without replacement' changes every number after the first stage.

3Two-way (contingency) tables

A contingency table cross-classifies two things. Always work out the row, column and grand totals first — most of the questions are answered straight off them.

	Bus	Walk	Total
Male	42	18	60
Female	28	12	40
Total	70	30	100

Worked ExampleAre 'male' and 'bus' independent in the table above?

1
P(male) = 60100 = 0,6.
Row total over grand total.
2
P(bus) = 70100 = 0,7.
Column total over grand total.
3
P(male and bus) = 42100 = 0,42.
The cell where they meet.
4
P(male) × P(bus) = 0,6 × 0,7 = 0,42.
5
0,42 = 0,42, so the events are independent.
State the conclusion in words — that is the mark.

💡

The independence test is always the same: compare P(A and B) with P(A) × P(B). Equal means independent; not equal means dependent.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
A bag holds 5 red and 3 blue beads. Two are drawn without replacement. Draw the tree diagram and calculate P(both red).
Show answer ▾Hide answer ▴
58 × 47 = 514
2
Using the tree diagram below, calculate P(one of each colour).
Show answer ▾Hide answer ▴
1556 + 1556 = 1528
3
Using the tree diagram below, calculate P(both blue).
Show answer ▾Hide answer ▴
38 × 27 = 656 = 328
4
If the two beads were drawn WITH replacement, calculate P(both red).
Show answer ▾Hide answer ▴
58 × 58 = 2564
5
In the two-way table (Male: 42 bus, 18 walk; Female: 28 bus, 12 walk), calculate P(female).
Show answer ▾Hide answer ▴
40100 = 0,4
6
For the same table, calculate P(walks to school).
Show answer ▾Hide answer ▴
30100 = 0,3
7
For the same table, determine whether 'male' and 'bus' are independent.
Show answer ▾Hide answer ▴
P(M)×P(B) = 0,6×0,7 = 0,42 and P(M and B) = 0,42, so yes, they are independent.
8
State the rule for multiplying along a tree and adding between paths.
Show answer ▾Hide answer ▴
Multiply along a path (AND); add between paths (OR).