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  • Need solution for RD Sharma Maths Class12 Chapter 30 Probability Exercise Fill in the Blanks Question, Question 15.

Need solution for RD Sharma Maths Class12 Chapter 30 Probability Exercise Fill in the Blanks Question, Question 15.

Answers (1)

Answer:  P(A)+P(B)

Given:

Ret A1B,C be pair wise independent events with P(C)>0
and P(A\cap B\cap C)=0. If P\left(\frac{A \cap B}{C}\right)=1-x, then x=…

Hint:

Using P\left(\frac{A}{B}\right)=\frac{P(A \cap B)}{P(B)}

Explanation:

Let A,B,C be pair wise independent events.
\begin{aligned} &\Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{C})=\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{C}) \\ &\mathrm{P}(\mathrm{B} \cap \mathrm{C})=\mathrm{P}(\mathrm{B}) \cdot \mathrm{P}(\mathrm{C})-(*) \end{aligned}

Given
So    P(A\cap B\cap C)=0                          

\begin{aligned} &\mathrm{P}(\bar{A} \cap \overline{\mathrm{B}} \cap \mathrm{C})=0 \\ &P\left(\frac{\overline{\mathrm{A}} \overline{\mathrm{B}}}{\mathrm{C}}\right)=\frac{\mathrm{P}[(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}) \cap \mathrm{C}]}{\mathrm{P}(\mathrm{C})} \\ &=\frac{\mathrm{P}(\mathrm{C})-\mathrm{P}(\mathrm{A} \cap \mathrm{C})-\mathrm{P}(\mathrm{B} \cap \mathrm{C})+\mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})}{\mathrm{P}(\mathrm{C})} \\ &=\frac{\mathrm{P}(\mathrm{C})-\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{C})-\mathrm{P}(\mathrm{B}) \mathrm{P}(\mathrm{C})+0}{\mathrm{P}(\mathrm{C})} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad[\because \text { Of }(*)] \end{aligned}

\begin{aligned} &=\frac{P(C) \times[1-P(A)-P(B)]}{P(C)} \\ &=1-P(A)-P(B) \\ &=1-[P(A)+P(B)]-(1) \end{aligned}

Also,

\mathrm{P}\left(\frac{\mathrm{A} \cap \mathrm{B}}{\mathrm{c}}\right)=1-\mathrm{x}-(2)

From (1) and (2),
We get x=P(A)+P(B)

Posted by

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