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  • Explain Solution R.D.Sharma Class 12 Chapter 30 Probability Exercise 30.4 Question 25 Sub Question 3 Maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter 30 Probability  Exercise 30.4 Question 25 Sub Question 3 Maths Textbook Solution.

Answers (1)

Answer: 1-\left ( 1-P_{1} \right )\left ( 1-P_{2} \right )=P\left ( Atleast\: one\: of\: A\: and\: B\: occurs \right )

Given: Let A and B be two independent events such that

             P\left ( A \right )=p_{1}\&P\left ( B \right )=p_{2}

Hint: Use,  P\left ( \bar{A}\right )\times P\left ( \bar{B} \right )=P\left ( \bar{A}\cap \bar{B} \right )

Solution:

 As we know, Two events are said to be independent if the product of the events are equal to their intersection. i.e. P\left ( A\cap B \right )=P\left ( A \right )P\left ( B \right )

Given, A and B are independent events

As, \begin{aligned} 1-\left(1-P_{1}\right)\left(1-P_{2}\right) &=1-[1-P(A)][1-P(B)] \\ \end{aligned}

                                                 \begin{aligned} &=1-P(\bar{A}) \times P(\bar{B}) \end{aligned}

And A and B are independent vectors

            \begin{aligned} &\mathrm{P}(\overline{\mathrm{A}}) \times \mathrm{P}(\overline{\mathrm{B}})=\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}) \\ &\begin{aligned} \Rightarrow 1-\left(1-\mathrm{P}_{1}\right)\left(1-\mathrm{P}_{2}\right)=& 1-\mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}) \\ &=1-\mathrm{P}(\overline{\mathrm{A} \cup \mathrm{B}}) \\ &=\mathrm{P}(\mathrm{A} \cup \mathrm{B}) \end{aligned} \\ \end{aligned}

So,\begin{aligned} &P(\mathrm{~A} \cup \mathrm{B})=1-\left(1-\mathrm{P}_{1}\right)\left(1-\mathrm{P}_{2}\right) \end{aligned}

Hence,1-\left ( 1-P_{1} \right )\left ( 1-P_{2} \right )=P\left ( Atleast\: one\: of\: A\: and\: B\: occurs \right )

            

                                                

So, 

Hence, 

Posted by

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