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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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A)
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• 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($$\bar{A}/\bar{B}$$)=$$\large\frac{5}{12}\times$$$$\large\frac{3}{2}$$
=$$\large\frac{15}{24}$$