Two-way tables and conditional probability

Reading a two-way table

Probability measures how likely something is, on a scale from 0 (impossible) to 1 (certain).
A two-way table classifies data by two variables at once (here, whether students studied and whether they passed).
Each inner cell counts people in both categories (42 students studied and passed); the margins give row totals, column totals, and the grand total (bottom-right).
To find a probability, divide the relevant count by the relevant total — choosing the right total is the key skill.

	Passed	Failed	Total
Studied	42	8	50
Did not study	15	35	50
Total	57	43	100

Conditional probability

A conditional probability is the chance of one event given that another has already happened.
It is written $P(A \mid B)$ — "the probability of $A$ given $B$ ".
The words "given", "of those who…" or "among" tell you to restrict to that group, so its total becomes the denominator — not the grand total.

$P(\text{passed} \mid \text{studied}) = \frac{42}{50} = 0.84$

Here the group is "studied" (50 students), and 42 of them passed.

Worked ExampleProbability from the grand total

Using the table (100 students), find the probability that a randomly chosen student passed.

Step 1 — No restriction, so use the grand total

"A randomly chosen student" means everyone, so the denominator is the grand total, 100.

Step 2 — Count the passers and divide

The Total column shows 57 students passed:

$P(\text{pass}) = \frac{57}{100} = 0.57$

Worked ExampleComparing conditional probabilities

Using the table above, compare the probability of passing for students who studied with those who did not.

Step 1 — Passing given studied

Restrict to the 50 who studied:

$P(\text{pass} \mid \text{studied}) = \frac{42}{50} = 0.84$

Step 2 — Passing given did not study

Restrict to the 50 who did not study:

$P(\text{pass} \mid \text{did not study}) = \frac{15}{50} = 0.30$

Step 3 — Interpret in context

Studying is associated with a much higher chance of passing (0.84 vs 0.30) — students who studied were nearly three times as likely to pass.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Using the table (100 students), find the probability that a randomly chosen student passed.

Merit

Using the table, find the probability that a student failed, given that they did not study.

Excellence

A student claims: 'Studying makes no difference to whether you pass.' Using the table, evaluate this claim.

Two-way tables and conditional probability

Covers AS91267

Reading a two-way table

Probability measures how likely something is, on a scale from 0 (impossible) to 1 (certain).
A two-way table classifies data by two variables at once (here, whether students studied and whether they passed).
Each inner cell counts people in both categories (42 students studied and passed); the margins give row totals, column totals, and the grand total (bottom-right).
To find a probability, divide the relevant count by the relevant total — choosing the right total is the key skill.

	Passed	Failed	Total
Studied	42	8	50
Did not study	15	35	50
Total	57	43	100

Conditional probability

A conditional probability is the chance of one event given that another has already happened.
It is written $P(A \mid B)$ — "the probability of $A$ given $B$ ".
The words "given", "of those who…" or "among" tell you to restrict to that group, so its total becomes the denominator — not the grand total.

$P(\text{passed} \mid \text{studied}) = \frac{42}{50} = 0.84$

Here the group is "studied" (50 students), and 42 of them passed.

Worked ExampleProbability from the grand total

Using the table (100 students), find the probability that a randomly chosen student passed.

Step 1 — No restriction, so use the grand total

"A randomly chosen student" means everyone, so the denominator is the grand total, 100.

Step 2 — Count the passers and divide

The Total column shows 57 students passed:

$P(\text{pass}) = \frac{57}{100} = 0.57$

Worked ExampleComparing conditional probabilities

Using the table above, compare the probability of passing for students who studied with those who did not.

Step 1 — Passing given studied

Restrict to the 50 who studied:

$P(\text{pass} \mid \text{studied}) = \frac{42}{50} = 0.84$

Step 2 — Passing given did not study

Restrict to the 50 who did not study:

$P(\text{pass} \mid \text{did not study}) = \frac{15}{50} = 0.30$

Step 3 — Interpret in context

Studying is associated with a much higher chance of passing (0.84 vs 0.30) — students who studied were nearly three times as likely to pass.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Using the table (100 students), find the probability that a randomly chosen student passed.

Merit

Using the table, find the probability that a student failed, given that they did not study.

Excellence

A student claims: 'Studying makes no difference to whether you pass.' Using the table, evaluate this claim.