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  • Explain Solution R.D. Sharma Class 12 Chapter 30 Probability Exercise 30.4 Question 25 Sub Question 4 Maths Textbook Solution.

Explain Solution R.D. Sharma Class 12 Chapter 30 Probability  Exercise 30.4 Question 25 Sub Question 4 Maths Textbook Solution.

Answers (1)

Answer: \mathrm{P}_{1}+\mathrm{P}_{2}-2 \mathrm{P}_{1} \mathrm{P}_{2}=\mathrm{P}( Exactly\: \: one \: \: of \mathrm{A} \& \mathrm{~B} occurs )

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

            P(A)=p_{1} \& P(B)=p_{2}

 Hint: P(A) \times P(B)=P(A \cap B)

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(A \cap B)=P(A) P(B)

Given,

         \begin{aligned} &P_{1}+P_{2}-2 P_{1} P_{2}=\left(P_{1}-P_{1} P_{2}\right)+\left(P_{2}-P_{1} P_{2}\right) \\ \end{aligned}

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

 And A and B are independent events

            i.eP(A) \times P(B)=P(A \cap B)

 \begin{aligned} \Rightarrow \mathrm{P}_{1}+\mathrm{P}_{2}-2 \mathrm{P}_{1} \mathrm{P}_{2} &=[\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})]+[\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})] \\ &=\mathrm{P}(\text { only } \mathrm{A})+\mathrm{P}(\text { only } \mathrm{B}) \end{aligned}

So, \mathrm{P}( only \mathrm{A})+\mathrm{P}( only mathrm{B})=\mathrm{P}_{1}+\mathrm{P}_{2}-2 \mathrm{P}_{1} \mathrm{P}_{2}                          

Hence, \mathrm{P}_{1}+\mathrm{P}_{2}-2 \mathrm{P}_{1} \mathrm{P}_{2}=\mathrm{P}( Exactly\: \: one \: \: of \mathrm{A} \& \mathrm{~B} occurs )

           

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