# Three events A,B and C have probabilities $\Large \frac{2}{5},\frac{1}{3}$ and $\Large \frac{1}{2}$,respectively.Given that $P(A\cap C)=\frac{1}{5}\;and\;P(B\cap C)=\frac{1}{4}$,find the value of $P(C|B)\; and \;P(A'\cap C').$

Toolbox:
• P(C/B)=P$$\frac{(B\cap\;C)}{P(B)}$$
• P($$\bar{A}\cap\bar{C}$$)=P($$\overline{A\cup\;C})$$
• =1-P(A$$\cup$$C)
• AlsoP(A$$\cup$$C)=P(A)+P(C)-P(A$$\cap$$C)
P(A)=$$\large\frac{2}{3}$$
P(B)=$$\large\frac{1}{3}$$
P(C)=$$\large\frac{1}{2}$$
P(A$$\cap$$C)=$$\large\frac{1}{5}$$
P(B$$\cap$$C)=$$\large\frac{1}{7}$$
P(C/B)=$$\large\frac{1}{7}\times$$$$\large\frac{3}{1}$$=$$\large\frac{3}{7}$$
P(A$$\cup$$C)=$$\large\frac{2}{3}$$+$$\large\frac{1}{2}$$-$$\large\frac{1}{5}$$
=$$\large\frac{29}{30}$$
P(A$$\cap$$C)=1-$$\large\frac{29}{30}$$=$$\large\frac{1}{30}$$

edited Jun 4, 2013