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Home  >>  CBSE XII  >>  Math  >>  Probability
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A and B are two events such that $P(A)=\frac{1}{2},P(B)=\frac{1}{3} and\; P(A\cap B)=\frac{1}{4}$.Find:\[(i)\;P(A|B)\quad(ii)\;P(B|A)\quad(iii)\;P(A'|B)\quad(iv)\;P(A'|B')\]

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1 Answer

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Toolbox:
  • P(A/B)=\(P\Large\frac{(A\cap\;B)}{P(B)}\)
  • P(B/A)=\(P\large\frac{(A\cap\;B)}{P(A)}\)
  • P(\(\bar{A}\)/B)=\(P\large\frac{(\bar{A}\cap\;B)}{P(B)}\)
  • P(\(\bar{A}\)/\(\bar{B}\))=\(P\large\frac{(A\cap\;\bar{B})}{P(B)}\)
  • P(\(\bar{B}\))=1-P(B)
  • \(P\;(\bar{A}/B)=P(B)-P(A\;\cap\;B)\)
  • P(\(\bar{A}\cap\;\bar{B}\))=P(\(\overline{A\;\cup\;B}\))=\(1-P(A\;\cup\;B)\)
  • \(P(A\;\cup\;B)=P(A)+P(B)-P(A\;\cap\;B)\)
P(A)=\(\large\frac{1}{2}\) P(B)=\(\large\frac{1}{3}\)
\(P(A\;\cap\;B)\)=\(\large\frac{1}{4}\)
1=P(A/B)=\(\large\frac{1}{4}\times\)\(\large\frac{3}{1}\)
=\(\large\frac{3}{4}\)
2=P(B/A)=\(\large\frac{1}{4}\times\)\(\large\frac{2}{1}\)
=\(\large\frac{1}{2}\)
3=\(P(\bar{A}\cap\;B\))=\(\large\frac{1}{3}\)-\(\large\frac{1}{4}\)=\(\large\frac{1}{12}\)
P(\(\bar{A}/B\))=\(\large\frac{1}{12}\times\)\(\large\frac{3}{1}\)=\(\large\frac{1}{4}\)
\(4=P(A\;\cup\;B)\)=\(\large\frac{1}{2}\)+\(\large\frac{1}{3}\)-\(\large\frac{1}{4}\)
\(P(\bar{A}\cap\;\bar{B})\)=\(1-\;P(A\;\cup\;B)\)
=1-\(\large\frac{7}{12}\)=\(\large\frac{5}{12}\)
P(\(\bar{A}/\bar{B}\))=\(\large\frac{5}{12}\times\)\(\large\frac{3}{2}\)
=\(\large\frac{15}{24}\)

 

answered Feb 25, 2013 by poojasapani_1
edited Jun 5, 2013 by poojasapani_1
 

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