Given That Events A and B Are Such That P(A) = 1/2, P(B) = p and P(A ∪ B) = 3/5, Find p if A and B Are Mutually Exclusive, and if A and B Are Independent

NCERT Class 12 Mathematics — Probability, Chapter 13.

Problem

Given that events $A$ and $B$ are such that $P(A)=\dfrac12$, $P(B)=p$, $P(A\cup B)=\dfrac35$. Find $p$ if (i) $A$ and $B$ are mutually exclusive, (ii) $A$ and $B$ are independent.

Key insight. Both cases start from the same addition rule, $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ — the only thing that changes between the two parts is what $P(A\cap B)$ is replaced with: zero for mutually exclusive events, or $P(A)P(B)$ for independent ones.

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Approach

Start from the general addition rule for two events, $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. For part (i), mutually exclusive events never occur together, so $P(A\cap B)=0$, turning the rule into a simple sum. For part (ii), independent events satisfy $P(A\cap B)=P(A)P(B)$, which substitutes a $p$-dependent term into the same rule — giving a slightly different, but still linear, equation to solve for $p$.

Solution

Step 1 — Mutually exclusive case: set up the equation

For mutually exclusive events, $P(A\cap B)=0$, so:

$$P(A\cup B) = P(A)+P(B)$$

$$\frac35 = \frac12+p$$

Step 2 — Solve for p

$$p = \frac35-\frac12 = \frac6{10}-\frac5{10} = \frac1{10}$$

Step 3 — Independent case: set up the equation

For independent events, $P(A\cap B)=P(A)P(B)=\dfrac12p$, so:

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

$$\frac35 = \frac12+p-\frac12p$$

Step 4 — Simplify the right-hand side

$$\frac35 = \frac12+\frac12p$$

Step 5 — Solve for p

$$\frac12p = \frac35-\frac12 = \frac1{10}$$

$$p = \frac2{10} = \frac15$$

Answer

Common mistakes

• Using the same equation for both parts. It’s a common shortcut error to solve part (i)’s simpler equation and then just reuse that value of $p$ for part (ii) — the two scenarios genuinely give different values of $p$, since independent events (unless one has probability 0) generally have some overlap, while mutually exclusive events have none.
• Forgetting to combine the p terms in Step 4. $p-\dfrac12p$ simplifies to $\dfrac12p$, not $p$ or $-\dfrac12p$ — dropping this simplification step risks an arithmetic slip in the next line.
• Assuming mutually exclusive and independent are compatible. For $P(A)=\dfrac12\ne0$ and $P(B)=\dfrac1{10}\ne0$, these two events genuinely cannot be both mutually exclusive and independent at once — the two parts of this question describe two entirely different (hypothetical) scenarios for $B$, not two properties of the same pair of events.

Practise next

• Given that events $A$ and $B$ are such that $P(A)=\dfrac25$, $P(B)=p$, $P(A\cup B)=\dfrac7{10}$. Find $p$ if (i) $A$ and $B$ are mutually exclusive, (ii) $A$ and $B$ are independent.