Year 11 Methods (Unit 1 & 2) Probability

Probability Tables

Q: How do you read a probability from a two-way table?

A joint probability is a single inside cell. A marginal probability is a row or column total. For counts, divide the cell by the grand total.

Q: How do you turn a frequency table into probabilities?

Divide each count by the grand total. The probability equals the cell count divided by N.

Q: What is a marginal probability?

A row or column total. For example the probability of A equals the probability of A and B plus the probability of A and not B.

Q: How do you find P(A or B) from a table?

Use the addition rule, the probability of A plus the probability of B minus the probability of A and B. Or take one minus the neither cell.

Q: What is the difference between a joint and a marginal probability?

A joint probability is a cell for both events. A marginal probability is a total for a single event.

20 practice questions 1 video lesson Theory + worked examples

Master two-way probability tables for Queensland Year 11 Mathematical Methods (QCAA). A two-way table sorts outcomes by two features, laying out counts so probabilities can be read at a glance.

You will learn to complete a table from row and column totals, tell joint from marginal probabilities, convert counts to probabilities using relative frequency, and apply the addition and complement rules — a practical skill for QCAA data and probability.

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Theory

In Year 11 Mathematical Methods (QCAA, Unit 1 Topic 5), a two-way table lays out events in rows and columns. This page shows how to read joint and marginal probabilities, complete a table from given totals, turn counts into probabilities (relative frequency), and apply the addition and complement rules from a table.

A two-way (contingency) table splits outcomes by two variables. The inside cells are the joint values \(A\cap B\), \(A\cap B'\), \(A'\cap B\), \(A'\cap B'\); the margins (row and column totals) give \(P(A)\), \(P(B)\) and their complements.

In a probability table every cell is a probability and the grand total is \(1\). In a frequency table the cells are counts and the grand total is the sample size \(N\); a probability is then the relative frequency \(\dfrac{\text{cell count}}{N}\).

Rows add across to the row total and columns add down to the column total, so any one missing value can be found by back-solving.

Everything adds up. Each row and column sums to its margin, the margins sum to the grand total, and the grand total is \(1\) (probabilities) or \(N\) (counts).

A completed two-way probability table (all cells add to 1)
	\(B\)	\(B'\)	Total
\(A\)	0.20	0.30	0.50
\(A'\)	0.15	0.35	0.50
Total	0.35	0.65	1

Read a joint probability straight from a cell: \(P(A\cap B)=0.20\). The margins give \(P(A)=0.50\) and \(P(B)=0.35\).

The four cells are exactly the four Venn regions. \(P(A\cup B)\) is everything except the ‘neither’ cell \(0.35\), so \(P(A\cup B)=0.65\).

Relative frequency from a frequency table:

\[P(\text{cell})=\dfrac{\text{cell count}}{\text{grand total }N}\]

P = \frac{cell count}{N}

Marginal probability (a row or column total):

\[P(A)=P(A\cap B)+P(A\cap B')\]

P (A) = P (A \cap B) + P (A \cap B')

Addition and complement from a table:

\[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Tip: \(P(A\cup B)\) is the whole table minus the ‘neither’ cell: \(1-P(A'\cap B')\).

How to work with a two-way table

Label the rows and columns with the events and their complements.
Fill the margins: each row and column adds to its total, and everything adds to \(1\) (or \(N\)).
Back-solve any missing cell by subtracting from a known total.
Answer: read a joint value from a cell, a marginal from a total, or apply the addition rule; for counts, divide by \(N\) to get a probability.

Example 1 — Counts to a probability

The table shows a survey of \(200\) people. Find the probability a randomly chosen person is female and owns a pet.

Survey of 200 people
	Owns a pet	No pet	Total
Male	60	40	100
Female	70	30	100
Total	130	70	200

Solution

Read the ‘female and owns a pet’ cell and the grand total:

\(\text{count}\)	\(=\)	\(70\)
\(N\)	\(=\)	\(200\)

Relative frequency \(=\dfrac{\text{cell}}{N}\):

\(P(\text{female}\cap\text{pet})\)	\(=\)	\(\dfrac{70}{200}\)
	\(=\)	\(\dfrac{7}{20}\)
	\(=\)	\(0.35\)

\(P(\text{female and pet})=\dfrac{7}{20}=0.35\).

Example 2 — Completing a table

Complete the probability table, then find \(P(B)\).

Fill in the missing probabilities
	\(B\)	\(B'\)	Total
\(A\)	0.20	0.30	?
\(A'\)	0.15	?	?
Total	?	?	1

Solution

The whole table adds to \(1\), so the missing bottom-right cell is:

\(P(A'\cap B')\)	\(=\)	\(1-(0.20+0.30+0.15)\)
	\(=\)	\(1-0.65\)
	\(=\)	\(0.35\)

Column \(B\) total is the marginal \(P(B)\):

\(P(B)\)	\(=\)	\(0.20+0.15\)
	\(=\)	\(0.35\)

\(P(A'\cap B')=0.35\) and \(P(B)=0.35\).

Example 3 — Addition rule from a table

Using the completed table below, find \(P(A\cup B)\).

The completed probability table
	\(B\)	\(B'\)	Total
\(A\)	0.20	0.30	0.50
\(A'\)	0.15	0.35	0.50
Total	0.35	0.65	1

Solution

Marginals from the totals:

\(P(A)\)	\(=\)	\(0.50\)
\(P(B)\)	\(=\)	\(0.35\)
\(P(A\cap B)\)	\(=\)	\(0.20\)

Apply the addition rule:

\(P(A\cup B)\)	\(=\)	\(0.50+0.35-0.20\)
	\(=\)	\(0.65\)

Check with the complement of the ‘neither’ cell:

\(1-P(A'\cap B')\)	\(=\)	\(1-0.35\)
	\(=\)	\(0.65\)

\(P(A\cup B)=0.65\).

Example 4 — Build a table and back-solve

Of \(80\) students, \(50\) study Maths, \(35\) study Physics and \(20\) study both. Build a two-way table, find how many study neither, and \(P(\text{Maths only})\).

80 students by subject
	Physics	No physics	Total
Maths	20	30	50
No maths	15	15	30
Total	35	45	80

Solution

Both is the overlap; Maths only \(=50-20\), Physics only \(=35-20\):

\(\text{Maths only}\)	\(=\)	\(50-20=30\)
\(\text{Physics only}\)	\(=\)	\(35-20=15\)

Neither \(=80-(\text{Maths}+\text{Physics}-\text{both})\):

\(\text{neither}\)	\(=\)	\(80-(50+35-20)\)
	\(=\)	\(80-65\)
	\(=\)	\(15\)

Probability of Maths only:

\(P(\text{Maths only})\)	\(=\)	\(\dfrac{30}{80}\)
	\(=\)	\(\dfrac{3}{8}\)

\(15\) study neither, and \(P(\text{Maths only})=\dfrac{3}{8}\).

Common pitfalls

Dividing a count by the wrong total. A probability is the cell count over the grand total \(N\), not over a row or column total (that would be conditional).

Mixing counts and probabilities. Decide whether the table adds to \(1\) (probabilities) or to \(N\) (counts) before you start.

Adding ‘both’ twice. When building a table, ‘Maths only’ \(=n(\text{Maths})-n(\text{both})\); do not leave the overlap in the ‘only’ cell.

Forgetting a margin must match. Every row and column has to add to its total; use this to check your completed table.

Frequently asked questions

How do you read a probability from a two-way table?

A joint probability is a single inside cell; a marginal probability is a row or column total. For counts, divide the cell by the grand total \(N\).

How do you turn a frequency table into probabilities?

Divide each count by the grand total: \(P=\dfrac{\text{cell count}}{N}\).

What is a marginal probability?

A row or column total, for example \(P(A)=P(A\cap B)+P(A\cap B')\).

How do you complete a missing cell?

Subtract the known values from the relevant total, since each row and column must add to its margin and the whole table to \(1\).

How do you find P(A or B) from a table?

Use \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), or take \(1\) minus the ‘neither’ cell \(P(A'\cap B')\).

What is the difference between a joint and a marginal probability?

A joint probability is a cell (both events); a marginal probability is a total for a single event.

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How to work with a two-way table

Common pitfalls

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