Question

1AandBaretwo events suchthat P(A)÷ P(A1B)-큼 and P(AIB)- Show that P(An B)-- 64 (10 marks)

Answer 1

By the law of total probability we have

$P(A)=P(A|B)P(B)+P(A|B')P(B')=P(A|B)P(B)+P(A|B')[1-P(B)]$

Putting values in the above equation gives

$\frac{1}{4}=\frac{3}{5}\cdot P(B)+\frac{1}{7}\cdot [1-P(B)]$

$\frac{1}{4}=\frac{16}{35}\cdot P(B)+\frac{1}{7}$

$\frac{1}{4}-\frac{1}{7}=\frac{16}{35}\cdot P(B)$

$\frac{16}{35}\cdot P(B)=\frac{3}{28}$

$P(B)=\frac{3\cdot 35}{28\cdot 16}=\frac{15}{64}$

So,

$P(A\cap B)=P(A|B)P(B)=\frac{3}{5}\cdot \frac{15}{64}=\frac{9}{64}$

Hence, proved