1
$\begingroup$

Assume $$P(B|A) = 1/5,\ P(A) = 3/4,\ P(A \cap B) = 3/20,\ \textrm{and}\ P(¬B|¬A) = 4/7.$$ Find $P(B)$.

What I tried: $$P(B)=\dfrac{ P(A \cap B)}{P(B|A)}=(3/20)/(1/5) = 3/4.$$

Answer is $P(B)=9/35.$

Where have I made the mistake?

$\endgroup$
  • $\begingroup$ Since P(B|A) = P(A,B) / P(A), how would you expect P(A,B) / P(B|A) to be P(B) ..? You found the answer, but to a different question. $\endgroup$ – Tim Sep 17 '18 at 15:44
  • $\begingroup$ @Tim Yes, how could I have overlooked that. I missed an important detail of the question - have edited it now. $\endgroup$ – user218970 Sep 17 '18 at 16:01
  • $\begingroup$ Same comment applies. $\endgroup$ – Tim Sep 17 '18 at 16:08
  • $\begingroup$ Could you post the exact question? $\endgroup$ – Tim Sep 18 '18 at 5:01
1
$\begingroup$

The probability of B can be split into the probability given A and given not A $$P(B) = P(B|A) \cdot P(A) + P(B|\neg A) \cdot P(\neg A)$$ The negations can be replaced by one minus the actual and vice versa $$P(B) = \frac{1}{5} \cdot \frac{3}{4} + (1-P(\neg B| \neg A)) \cdot (1-P(A))$$ $$P(B) = \frac{3}{20} + (1-\frac{4}{7})\cdot (1-\frac{3}{4})$$ $$P(B) = \frac{3}{20} + \frac{3}{7} \cdot \frac{1}{4}$$ $$p(B) = \frac{3}{20} + \frac{3}{28}$$ $$p(B) = \frac{21}{140} + \frac{15}{140}$$ $$P(B) = \frac{36}{140}$$ $$P(B) = \frac{9}{35}$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy