Types of events in probability — mutually exclusive, independent, exhaustive

Question

Two dice are thrown. Let A = “sum is 7” and B = “first die shows 3”. Are events A and B (a) mutually exclusive? (b) independent? Explain with calculations.

(CBSE Class 10/11/12 pattern)

Solution — Step by Step

Total outcomes when two dice are thrown: $6 \times 6 = 36$ .

Event A (sum is 7): $\{(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\}$ → $P(A) = 6/36 = 1/6$

Event B (first die shows 3): $\{(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)\}$ → $P(B) = 6/36 = 1/6$

flowchart TD
    A["Event Classification"] --> B{"A ∩ B = ∅ ?"}
    B -->|"Yes"| C["Mutually Exclusive\nP(A ∩ B) = 0\nCannot happen together"]
    B -->|"No"| D["Not Mutually Exclusive\nP(A ∩ B) > 0\nCan happen together"]
    A --> E{"P(A ∩ B) = P(A)×P(B)?"}
    E -->|"Yes"| F["Independent\nOne event doesn't\naffect the other"]
    E -->|"No"| G["Dependent\nKnowing one changes\nprobability of other"]

$A \cap B$ = outcomes where sum is 7 AND first die is 3 = $\{(3,4)\}$ .

Since $A \cap B \neq \emptyset$ , events A and B are not mutually exclusive. They can occur simultaneously.

$P(A \cap B) = 1/36$

$P(A) \times P(B) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$

Since $P(A \cap B) = P(A) \times P(B)$ , events A and B are independent.

Knowing that the first die shows 3 does not change the probability of the sum being 7. Why? Because regardless of the first die value, exactly one out of six second-die values gives sum = 7.

Why This Works

Mutually exclusive and independent are two completely different concepts. Mutually exclusive means the events cannot both happen — they share no outcomes. Independent means the occurrence of one does not affect the probability of the other.

A crucial insight: mutually exclusive events are never independent (unless one has probability zero). If $A \cap B = \emptyset$ , then knowing A occurred tells you B definitely did not — that is maximum dependence, not independence.

Alternative Method — Venn Diagram Check

Draw two circles in the sample space. If the circles do not overlap → mutually exclusive. If the area of overlap equals the product of the two areas (relative to total) → independent. Both conditions are rarely satisfied simultaneously.

For CBSE Class 12, the addition theorem connects these: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ . For mutually exclusive events, this simplifies to $P(A \cup B) = P(A) + P(B)$ . For independent events, $P(A \cap B) = P(A) \cdot P(B)$ . These two formulas cover most board exam problems.

Common Mistake

The most dangerous confusion: treating “mutually exclusive” and “independent” as the same thing. Students write $P(A \cap B) = 0$ for independent events, or $P(A \cap B) = P(A) \cdot P(B)$ for mutually exclusive events. These are opposite conditions. If events are mutually exclusive with non-zero probabilities, $P(A \cap B) = 0 \neq P(A) \cdot P(B)$ , so they are dependent. Always check the definitions separately.

Question

Solution — Step by Step

Why This Works

Alternative Method — Venn Diagram Check

Common Mistake

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