Question
$\text{If}\ \text{P}(\text{A})=\frac{6}{11},\ \text{P}(\text{B})=\frac{5}{11}\ \text{and}\ \text{P}(\text{A}\cup\text{B})=\frac{7}{11},$find
$\text{P}(\text{A}\cap\text{B})$
$\text{P}(\text{A}\cap\text{B})$
✓
Answer
$\text{P}\left(\text{A}\cup\text{B}\right)= \text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)-\text{P}\left(\text{A}\cap\text{B}\right)$
$\Rightarrow\ \ \ \frac {7}{11}=\frac{6}{11}+\frac{5}{11}-\text{P}\left(\text{A}\cap\text{B}\right)$
$ \Rightarrow\ \ \ \text {P}\left(\text{A}\cap\text{B}\right)=\frac{6}{11}+\frac{5}{11}-\frac{7}{11}=\frac{4}{11}$
$\Rightarrow\ \ \ \frac {7}{11}=\frac{6}{11}+\frac{5}{11}-\text{P}\left(\text{A}\cap\text{B}\right)$
$ \Rightarrow\ \ \ \text {P}\left(\text{A}\cap\text{B}\right)=\frac{6}{11}+\frac{5}{11}-\frac{7}{11}=\frac{4}{11}$
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