# If $$P(A) = \large \frac{6}{11}$$$$, P(B) =\large \frac{5}{11}\:$$ and $$P(A \cup B) = \large \frac{7}{11}$$ , find (i) P(A $$\cap$$ B) , (ii) P(A|B) , (iii) P(B|A)

$\begin{array}{1 1} \text{(i) 4/11, (ii) 2/11, (iii) 3/11} \\ \text{(i) 4/11, (ii) 4/5, (iii) 2/3} \\ \text{(i) 7/11, (ii) 3/5, (iii) 2/3} \\ \text{(i) 7/11, (ii) 3/5, (iii) 4/5} \end{array}$

Toolbox:
• $$1)\;p(A\cup\;B)=p(A)\;+\;p(B)\;-\;p(A\;\cap\;B)$$ $$\Rightarrow\;p(A\;\cap\; B)=p(A)\;+\;p(B)\;-\;p(A\;\cup\;B)$$
• $$2)\;p(A/B)=\large\frac{p(A\;\cap\;B)}{p(B)}$$
• $$3)\;p(B/A)=\large\frac{p(A\;\cap\;B)}{p(A)}$$
Given $p(A) = \large\frac{6}{11}$ and $p(B) = \large\frac{5}{11}$, $p(A\;\cup\;B) = \large\frac{7}{11}$
Therefore, $$p(A\cap\;B)=\large\frac{6}{11}\;+\;\frac{5}{11}\;-\;\frac{7}{11}\;=\;\frac{4}{11}$$
$$p(A/B)=\Large \frac{\frac{4}{11}}{\frac{5}{11}}$$ =$$\large\frac{4}{5}$$
$$p(B/A)=\Large \frac{\frac{4}{11}}{\frac{6}{11}}$$ =$$\large \frac{2}{3}$$
edited Jun 18, 2013