Learn8300Papers 1–3 · Higher tierProbabilityWithout replacement and Venn diagrams

Probability · Subtopic 3 of 4

Without Replacement and Venn Diagrams

When an item is not put back, the second pick has one fewer item, and possibly one fewer of the colour you took. A Venn diagram sorts a group into "only $A$ ", "both", "only $B$ " and "neither": fill in the overlap first.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Change the second-stage probabilities when items are not replaced.
Find "both the same" and "one of each" probabilities without replacement.
Complete a Venn diagram from given totals.
Read probabilities from a Venn diagram.

Pause and think

A bag has $2$ red and $1$ blue sweet. You eat one red sweet. What is the probability the next one is red?

Section 2 of 14

Core idea

Start hereSweets from a bag

Take a sweet and eat it, then take another. The bag now has one fewer sweet, so the second probabilities have changed. Events like this are dependent.

The big idea

Without replacement: second-stage totals drop by one. Conditional probability $P(A\mid B)$ : count only inside $B$ .

1. Without replacement

7 red and 3 blue counters: two taken without replacement

After a red is taken, 6 reds are left out of 9 counters.

P(\text{same colour})=\tfrac{42+6}{90}=\tfrac{8}{15}

Worked example7 red, 3 blue, take two

P(same colour), without and with replacement.

Goal: Compare.
Strategy: Tree; adjust totals.

$\tfrac7{10}\cdot\tfrac69=\tfrac{42}{90}$
Both red: one red gone.
$\tfrac3{10}\cdot\tfrac29=\tfrac6{90}$
Both blue.
$\tfrac{48}{90}=\tfrac8{15}$
Add.
$0.49+0.09=0.58$
With replacement.

Answer

\frac8{15}\approx0.533

;

0.58

✓ CheckWithout replacement, matching is harder ✓.

2. Venn diagrams

40 people: 25 have a cat, 18 a dog, 10 both

P(\text{dog}\mid\text{cat})

: only look inside the cat circle (25 people), of whom 10 have a dog:

\tfrac{10}{25}=\tfrac25

P(A\mid B)=\frac{n(A\cap B)}{n(B)}

$\mid$: "given".

Worked examplePets

40 people; 25 cat, 18 dog, 10 both.

Goal: P(dog only); P(dog | cat).
Strategy: Fill the Venn; restrict.

$15,\ 10,\ 8,\ 7$
Cat only, both, dog only, neither.
$P(\text{dog only})=\tfrac8{40}=\tfrac15$
Out of everyone.
$P(D\mid C)=\tfrac{10}{25}=\tfrac25$
Only among cat owners.

Answer

\frac15

;

\frac25

✓ Check

15+10+8+7=40

✓.

Common misconceptionWrong denominator For "given cat", divide by 25, not 40.

Pause and think

P(cat | dog)?

Summary

Reduce totals without replacement.
Conditional: restrict to the group.

Conditional probability: Probability given extra information.

HistoryJohn Venn. John Venn, an English logician, introduced his overlapping circles in 1880 to picture logical statements. A stained-glass window at Gonville and Caius College, Cambridge, where he studied and taught, shows a Venn diagram in his honour.

Fun factThe birthday surprise. In a group of just

23

people, the chance that two share a birthday is over

50\%

. Working it out uses a long chain of "without replacement" probabilities:

\tfrac{365}{365}\times\tfrac{364}{365}\times\tfrac{363}{365}\times\cdots

for no match.

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