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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:$(i)\;P(A|B)\quad(ii)\;P(B|A)\quad(iii)\;P(A'|B)\quad(iv)\;P(A'|B')$

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}$
=$\large\frac{15}{24}$

edited Jun 5, 2013