CBSE Class 12 Mathematics 2023: question 30(b)

Hints

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Step-by-step solution

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1. Since A and B are independent, so are A, $\bar{B}$ and $\bar{A}$, B. Hence $P(A)P(\bar{B})=\frac{1}{4}$ and $P(\bar{A})P(B)=\frac{1}{6}$.

   ½ mark

2. Let $P(A)=x$ and $P(B)=y$. Then $x(1-y)=\frac{1}{4}$ and $(1-x)y=\frac{1}{6}$.

   Subtracting: $x-y=\frac{1}{12}$, so $y=x-\frac{1}{12}$.

   1 mark

3. Substituting into the first equation: $x\left(\frac{13}{12}-x\right)=\frac{1}{4}$, which gives $12x^2-13x+3=0$.

   Factorising: $(3x-1)(4x-3)=0$, so $x=\frac{1}{3}$ or $\frac{3}{4}$.

   1 mark

4. If $P(A)=\frac{1}{3}$, then $P(B)=\frac{1}{4}$.

   If $P(A)=\frac{3}{4}$, then $P(B)=\frac{2}{3}$.

   ½ mark

Answer: $P(A)=\frac{1}{3},P(B)=\frac{1}{4}$ or $P(A)=\frac{3}{4},P(B)=\frac{2}{3}$

Revise the chapter

• Chapter: Probability (notes, formulas, where marks are lost and every past question)
• Topic: Independent Events