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  • Provide solution for RD Sharma Maths Class12 Chapter 30 Probability Exercise Fill in the Blanks Question, Question 1.

Provide solution for RD Sharma Maths Class12 Chapter 30 Probability Exercise Fill in the Blanks Question, Question 1.

Answers (1)

Answer:  P(A)P(\overline{B})

Given:

If A and B are independent events then P(A∪B)=1-x where x=…. -

Hint:

Using P(A)=1-P(A) ; P(A)=1-P(A)

Explanation:

We are given
A and \overline{B} an indepindent events
\Rightarrow \overline{A} and B oure also independent

Since \overline{A} and B are independent events.

\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B})=\mathrm{P}(\overline{\mathrm{A}}) \mathrm{P}(\mathrm{B})                                            

Now 

\begin{aligned} &\mathrm{P}(\overline{\mathrm{A}} \cup \mathrm{B})=\mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B}) \\ &=(1-\mathrm{P}(\mathrm{A}))+(1-\mathrm{P}(\overline{\mathrm{B}}))-\mathrm{P}(\overline{\mathrm{A}}) \mathrm{P}(\mathrm{B}) \\ &=(1-\mathrm{P}(\mathrm{A}))+(1-\mathrm{P}(\overline{\mathrm{B}}))-\cdot(1-\mathrm{P}(\mathrm{A}))(1-\mathrm{P}(\overline{\mathrm{B}})) \end{aligned}

\begin{aligned} &=1-\mathrm{P}(\mathrm{A})+1-\mathrm{P}(\overline{\mathrm{B}})-[1-\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{A}) \mathrm{P}(\overline{\mathrm{B}})] \\ &=1-\mathrm{P}(\mathrm{A})+1-\mathrm{P}(\overline{\mathrm{B}})-1+\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A})(\overline{\mathrm{B}}) \\ &=1-\mathrm{P}(\mathrm{A}) \mathrm{P}(\overline{\mathrm{B}}) \\ &\Rightarrow \mathrm{P}(\overline{\mathrm{A}} \cup \mathrm{B})=1-\mathrm{P}(\mathrm{A}) \mathrm{P}(\overline{\mathrm{B}})-(1) \end{aligned}

Also given that P(\overline{A}\cup B)=1-x-(2)

Comparing (1) and 2 we get x=1-P(A)P(\overline{B})

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