Independence & Mutual Exclusivity
Two events cannot both happen — are they independent? Many students say yes. They are wrong. "Mutually exclusive" and "independent" are the most confused pair in Module 5. By the end of this lesson you'll draw a bright line between them and never mix them up again.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
Consider two events $A$ and $B$. Can two events be both independent and mutually exclusive? Explain your reasoning before reading on — this is the core question of the whole lesson.
The entire lesson hangs on keeping two definitions distinct. Write them down now and refer back constantly.
Independence is about information — does knowing one event tells you nothing about the other. Mutual exclusivity is about possibility — can both events even happen at the same time?
Key facts
- Independence: $P(A \cap B) = P(A) \times P(B)$
- Mutual exclusivity: $P(A \cap B) = 0$
- Mutually exclusive events with $P > 0$ are always dependent
Concepts
- Independence = information: knowing one event tells you nothing about the other
- Mutual exclusivity = possibility: the events cannot coexist — maximally dependent
- The only way to be both: if one event has probability zero
Skills
- Test whether two events are independent using either formula
- Classify event pairs as independent, dependent, mutually exclusive, or none
- Critique real-world claims that misuse "independent"
Two events are independent if the occurrence of one does not affect the probability of the other.
Equivalently, using conditional probability:
What independence means intuitively: Knowing $B$ happened gives you absolutely no information about whether $A$ happened. The events are statistically unrelated.
Examples of independent events:
- Flipping a coin and rolling a die — the coin does not care about the die
- Two separate lottery draws — last week's numbers do not affect this week's
- Drawing with replacement — the first draw is forgotten before the second
Independent events have overlap — they can both happen. Mutually exclusive events have no overlap — they cannot both happen.
Testing for independence: Calculate $P(A) \times P(B)$ and compare to $P(A \cap B)$. If equal: independent. If not: dependent.
Independence test: $P(A \cap B) = P(A) \times P(B)$ — or equivalently $P(A \mid B) = P(A)$; Independent events can overlap — the Venn circles have an intersection
Pause — copy the two equivalent independence tests: $P(A \cap B) = P(A) \times P(B)$, or $P(A \mid B) = P(A)$, and the key insight that independent events CAN overlap (Venn circles do intersect) into your book.
Did you get this? True or false: if $A$ and $B$ are independent, then $P(A \cap B) = P(A) \times P(B)$.
Worked examples · 3 in a row, reveal as you go
A fair die is rolled. Let $A$ = "roll an even number" and $B$ = "roll a number greater than 4". (a) Are $A$ and $B$ mutually exclusive? (b) Are $A$ and $B$ independent?
In 200 students: 80 study French, 60 study German, 20 study both. Test whether studying French and studying German are independent. Are they mutually exclusive?
Two events have $P(A) = 0.5$, $P(B) = 0.5$, and $P(A \cap B) = 0$. Are they independent? Explain why mutually exclusive events with positive probability are always dependent.
Quick check: $P(A) = 0.3$, $P(B) = 0.4$, $P(A \cap B) = 0.12$. Which statement is correct?
Common errors · the traps that cost marks
Odd one out: Three of these statements are correct. Which one is the odd one out (the incorrect statement)?
Fill in the blank: If $P(A) > 0$ and $P(B) > 0$, and $A$ and $B$ are mutually exclusive, then $A$ and $B$ must be _____________.
Drill activities
A die is rolled. $A$ = "even number", $B$ = "prime number". Find $P(A)$, $P(B)$, $P(A \cap B)$. Are $A$ and $B$ independent?
A card is drawn. $C$ = "face card", $D$ = "red card". Are $C$ and $D$ independent? Are they mutually exclusive?
$P(A) = 0.3$, $P(B) = 0.4$, $P(A \cap B) = 0.12$. Are $A$ and $B$ independent?
$P(A) = 0.5$, $P(B) = 0.5$, $P(A \cap B) = 0$. Are $A$ and $B$ independent? Are they mutually exclusive?
Prove: if $A$ and $B$ are independent, then $A'$ and $B'$ are also independent. (Hint: use $P(A' \cap B') = 1 - P(A \cup B)$ and expand.)
Earlier you were asked: can two events be both independent and mutually exclusive? The answer: only if one event has probability zero. Here is why: if mutually exclusive, $P(A \cap B) = 0$. If also independent, $P(A \cap B) = P(A) \times P(B)$. So $P(A) \times P(B) = 0$, meaning at least one probability is zero. For any real events with $P(A) > 0$ and $P(B) > 0$, mutual exclusivity and independence are themselves mutually exclusive properties. Mutually exclusive events are maximally dependent: knowing one happened guarantees the other did not.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. A fair die is rolled. Let $A$ = "roll a multiple of 2" and $B$ = "roll a multiple of 3". (a) List the outcomes in $A$, $B$, and $A \cap B$. (b) Find $P(A)$, $P(B)$, and $P(A \cap B)$. (c) Determine whether $A$ and $B$ are independent, justifying with calculations. (d) Determine whether $A$ and $B$ are mutually exclusive. (3 marks)
Q2. In a survey of 200 students, 80 study French, 60 study German, and 20 study both. A student is selected at random. (a) Find $P(\text{French})$, $P(\text{German})$, and $P(\text{French} \cap \text{German})$. (b) Test whether studying French and studying German are independent events. (c) Are they mutually exclusive? Explain. (3 marks)
Q3. A newspaper headline reads: "Study Finds Two Cancer Risk Factors Are Completely Independent — Patients with One Factor Never Have the Other, So the Risks Do Not Interact." (a) Identify the mathematical error in this statement and explain why the journalist has confused two distinct probability concepts. (b) Rewrite the headline correctly for each of two possible intended meanings: (i) the risk factors cannot co-occur, and (ii) the presence of one risk factor does not change the probability of the other. (c) Explain why confusing these concepts in medical reporting could have serious public health consequences. (3 marks)
Comprehensive answers (click to reveal)
Drill 1: $A=\{2,4,6\}$, $B=\{2,3,5\}$, $A\cap B=\{2\}$. $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{2}$, $P(A\cap B)=\frac{1}{6}$. $P(A)\times P(B)=\frac{1}{4}\neq\frac{1}{6}$. Not independent.
Drill 2: $P(C)=\frac{12}{52}=\frac{3}{13}$, $P(D)=\frac{1}{2}$, $P(C\cap D)=\frac{6}{52}=\frac{3}{26}$. $P(C)\times P(D)=\frac{3}{26}=P(C\cap D)$. Independent. $P(C\cap D)\neq 0$: not mutually exclusive.
Drill 3: $P(A)\times P(B)=0.12=P(A\cap B)$. Independent.
Drill 4: ME (intersection = 0). $P(A)\times P(B)=0.25\neq 0$: not independent. Dependent.
Drill 5: $P(A'\cap B')=1-P(A\cup B)=1-[P(A)+P(B)-P(A)P(B)]=(1-P(A))(1-P(B))=P(A')P(B')$. So independent.
Q1 (3 marks): (a) $A=\{2,4,6\}$, $B=\{3,6\}$, $A\cap B=\{6\}$ [0.5]. (b) $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{3}$, $P(A\cap B)=\frac{1}{6}$ [0.5]. (c) $P(A)\times P(B)=\frac{1}{6}=P(A\cap B)$. Therefore independent [1]. (d) $P(A\cap B)=\frac{1}{6}\neq 0$, so not mutually exclusive [0.5+0.5].
Q2 (3 marks): (a) $P(F)=0.4$, $P(G)=0.3$, $P(F\cap G)=0.1$ [0.5]. (b) $P(F)\times P(G)=0.12\neq 0.1$. Therefore not independent (dependent) [1.5]. (c) $P(F\cap G)=0.1\neq 0$, so not mutually exclusive [0.5+0.5].
Q3 (3 marks): (a) The journalist confuses mutual exclusivity ($P(A\cap B)=0$, cannot co-occur) with independence (no statistical relationship). Mutually exclusive events are dependent, not independent [1]. (b)(i) "Study Finds Two Cancer Risk Factors Are Mutually Exclusive — Presence of One Rules Out the Other" [0.5]. (ii) "Study Finds Two Cancer Risk Factors Are Statistically Independent — Presence of One Does Not Affect Risk of the Other" [0.5]. (c) Conflating the concepts could lead patients to believe avoiding one risk factor automatically protects them from the other (if interpreted as independent), or that having one guarantees safety from the other (if interpreted as mutually exclusive). Both errors distort medical decision-making and public health messaging [1].
Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms using independence tests, mutual exclusivity, and event classification. Lighter alternative to the boss.
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