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Questions  >>  CBSE XII  >>  Math  >>  Probability
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Q)

If A and B are such that \[P(A'\cup B')=\frac{2}{3} \;and\;P(A\cup B)=\frac{5}{9},\]\[then\;P(A')+P(B')=\text{__________}.\]

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A)
Toolbox:
  • If $A$ and $B$ are two events associated with an experiment.
  • $P(\overline{A\cup B})=P(\bar{A}\cap \bar{B})\;$(1)
  • When $ P(\bar{A})=1-P(A)$
  • $P(\bar{B})=1-P(B)$
  • $P(A\cup B)=P(A)+P(B)-P(A\cap B)\;$(2)
$P(A'\cup B')=\large\frac{2}{3}$
$\Rightarrow P(\overline{ A\cap B})=\large\frac{2}{3}$
$1-P(A\cap B)=\large\frac{2}{3}$
$P(A\cap B)=1-\large\frac{2}{3}=\frac{1}{3}$
Also $P(A\cup B)=\large\frac{5}{9}$
$P(A)+P(B)-P(A\cap B)+P(A\cup B)$
$1-P(\bar{A})+1-P(A\cap B)=P(A\cup B)$
$P(\bar{A})+P(\bar{B})=2-P(A\cap B)=P(A\cup B)$
=$\Large 2-\frac{5}{9}-\frac{1}{3}=\frac{10}{9}$
$P(\bar{A})+P(\bar{B})=\large\frac{10}{9}$
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