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  • Provide solution for RD Sharma Maths Class12 Chapter 30 Probability Exercise Fill in the Blanks Question, Question 3.

Provide solution for RD Sharma Maths Class12 Chapter 30 Probability Exercise Fill in the Blanks Question, Question 3.

Answers (1)

Answer:   \frac{10}{9}

Given:

If A and B are two events such that P(\overline{A}\cup \overline{B})=\frac{2}{3}  and
P(A\cup B)=\frac{5}{9}

 Then P(\overline{A})+P(\overline{B})= ....

Hint:

Using P(\overline{A})=1-P(A)

Explanation:

Let A and B be the two events.

\begin{gathered} P(\bar{A} \cup \bar{B})=\frac{2}{3} \\ P(\overline{A \cap B})=\frac{2}{3} \\ \Rightarrow 1-P(A \cap B)=\frac{2}{3} \\ \Rightarrow P(A \cap B)=1-\frac{2}{3} \\ \Rightarrow P(A \cap B)=\frac{1}{3} \\ P(A \cup B)=\frac{5}{9} \text { (given) } \end{gathered}
Also ,

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

Now we know,

\begin{aligned} &\frac{5}{9}=2-(\mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}}))-\frac{1}{3} \\ &\Rightarrow \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=2-\frac{5}{9}-\frac{1}{3} \\ &\Rightarrow \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=\frac{18-5-3}{9}=\frac{10}{9} \\ &\Rightarrow \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})=\frac{10}{9} \end{aligned}

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