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Mathematics Question on Probability

Given that P(A)=0.1,P(B∣A)=0.6P(A) = 0.1, P(B | A) = 0.6 and P(B∣Ac)=0.3 P(B |A^c ) = 0.3 what is P(A∣B)P(A | B) ?

A

211\frac{2}{11}

B

411\frac{4}{11}

C

711\frac{7}{11}

D

911\frac{9}{11}

Answer

211\frac{2}{11}

Explanation

Solution

Given, P(B∣A)=0.6P( B|A) = 0.6
⇒P(B∩A)P(A)=0.6\Rightarrow \frac{P\left(B \cap A\right)}{P\left(A\right)} = 0.6
⇒P(B∩A)=0.6×0.1=0.06\Rightarrow P(B \cap A) = 0.6 \times 0.1 = 0.06
Also, P(B∣AC)=0.3P(B|A^C) = 0.3
⇒P(B∩AC)P(AC)=0.3\Rightarrow \frac{P\left(B \cap A^C\right)}{P\left(A^C\right)} = 0.3
⇒P(B)−P(B∩A)1−P(A)=0.3\Rightarrow \frac{P\left(B \right) - P \left(B \cap A\right)}{1 - P\left(A\right)} = 0.3
⇒P(B)−0.060.9=0.3\Rightarrow \frac{P\left(B \right) - 0.06 }{0.9 } = 0.3
⇒P(B)=0.33\Rightarrow P\left(B \right) = 0.33
Now, P(A∣B)=P(A ∩ B)P(B)=0.060.33=211P\left(A | B\right)=\frac{P\left(A \,\cap \,B\right)}{P\left(B\right)}=\frac{0.06}{0.33}=\frac{2}{11}