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Probability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsProbability1 Mark
Question
Fill in the blanks.
If A and B are such that $\text{P}(\text{A}'\cup\text{B}')=\frac{2}{3}$ and $\text{P}(\text{A}\cup\text{B})=\frac{5}{9},$ then $\text{P}(\text{A}')+\text{P}(\text{B}')=$ ________.
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Answer

If A and B are such that $\text{P}(\text{A}'\cup\text{B}')=\frac{2}{3}$ and $\text{P}(\text{A}\cup\text{B})=\frac{5}{9},$ then $\text{P}(\text{A}')+\text{P}(\text{B}')=\frac{10}{9}.$ Solution:
Here, $\text{P}(\text{A}'\cup\text{B}')=\frac{2}{3}$ and $\text{P}(\text{A}\cup\text{B})=\frac{5}{9}$
$\text{P}(\text{A}'\cup\text{B}')=1-\text{P}(\text{A}\cap\text{B})$
$\Rightarrow\text{P}(\text{A}\cap\text{B})=1-\frac{2}{3}=\frac{1}{3}$
$\because\text{P}(\text{A}')+\text{P}(\text{B}')=1-\text{P}(\text{A})+1-\text{P}(\text{B})$
$=2-\big[\text{P}(\text{A})+\text{P}(\text{B})\big]$
$=2-\big[\text{P}(\text{A}\cup\text{B})+\text{P}(\text{A}\cap\text{B})\big]$
$=2-\Big(\frac{5}{9}+\frac{1}{3}\Big)=\frac{10}{9}$

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