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  • Provide solution for RD Sharma Maths Class12 Chapter 30 Probability Exercise Fill in the Blanks Question, Question 2.

Provide solution for RD Sharma Maths Class12 Chapter 30 Probability Exercise Fill in the Blanks Question, Question 2.

Answers (1)

Answer:   \frac{5}{12},\frac{1}{3}

Given:

If A and B are independent events such-that

PA=p,

P(B)=2pand

P( Exactly one ofA_1B=\frac{5}{9}) then

p=?

Hint:

P(A \cap A B)=P(A) P(B) if A and B are independent.

Explanation:

As A and B ane independent events.
So P(A \cap A B)=P(A) P(B)       - (1)

Now
P( Exadly one of A and B occurs )= \frac{5}{9}
⇒P( only A)+P( onlyB)= \frac{5}{9}

\begin{aligned} &\Rightarrow[\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})]+[\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})]=\frac{5}{9} \\ &\Rightarrow[\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B})]+[\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B}))]=\frac{5}{9} \\ &\Rightarrow\left[\mathrm{P}(\mathrm{A}) \times(1-\mathrm{P}(\mathrm{B}))+\mathrm{P}(\mathrm{B}) \times[1-\mathrm{P}(\mathrm{A})]=\frac{5}{9}\right. \\ &\Rightarrow \mathrm{p}(1-2 \mathrm{p})+2 \mathrm{p}(1-\mathrm{p})=\frac{5}{9} \\ &\Rightarrow \mathrm{p}-2 \mathrm{p}^{2}+2 \mathrm{p}-2 \mathrm{p}^{2}=\frac{5}{9} \\ &\Rightarrow 3 \mathrm{p}-4 \mathrm{p}^{2}=\frac{5}{9} \\ &\Rightarrow 27 \mathrm{p}-36 \mathrm{p}^{2}=5 \\ &\Rightarrow 36 \mathrm{p}^{2}-27 \mathrm{p}+5=0 \end{aligned}

So using quadratic formula, \mu.

\begin{aligned} &p=\frac{-(-27) \pm \sqrt{(-27)^{2}-4 \times 36 \times 5}}{2 \times 36} \\ &=\frac{27 \pm \sqrt{729-720}}{72} \\ &=\frac{27 \pm \sqrt{9}}{72} \\ &=\frac{27 \pm 3}{72} \end{aligned}

\begin{aligned} p &=\frac{27+3}{72}, \frac{27-3}{72} \\ &=\frac{30}{72}, \frac{24}{72} \\ &=\frac{5}{12}, \text { or } \frac{1}{3} \end{aligned}

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