Question

If A and B are two events and P (A) = 2/3, P(B) =3/5, P AU   B

= 5/6, then the value of

  ' '

P A /B

is :​

Answer 1

Given : -

$\rm :\longmapsto\:P(A) = \dfrac{2}{3}$

$\rm :\longmapsto\:P(B) = \dfrac{3}{5}$

$\rm :\longmapsto\:P(A \: \cup \: B) = \dfrac{5}{6}$

To Find :-

$\rm :\longmapsto\:P(A |B)$

Solution :-

We know that,

$\rm :\longmapsto\:P(A \: \cup \:B) = P(A) + P(B) - P(A \: \cap \:B)$

On substituting the given values, we get

$\rm :\longmapsto \: \dfrac{5}{6} = \dfrac{2}{3} + \dfrac{3}{5} - P(A \: \cap \:B)$

$\rm :\longmapsto \:P(A \: \cap \:B) = \dfrac{2}{3} + \dfrac{3}{5} - \dfrac{5}{6}$

$\rm :\longmapsto \:P(A \: \cap \:B) = \dfrac{20 + 18 - 25}{30}$

$\rm :\longmapsto \:P(A \: \cap \:B) = \dfrac{38 - 25}{30}$

$\bf :\longmapsto \:P(A \: \cap \:B) = \dfrac{13}{30}$

Now,

Consider,

$\rm :\longmapsto\:P(A |B)$

$\rm \: = \: \dfrac{P(A \: \cap \:B)}{P(B)}$

$\rm \: = \: \dfrac{13}{30} \times \dfrac{5}{3}$

$\rm \: = \: \dfrac{13}{18}$

Hence,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \blue{\underbrace{\red{\boxed{\bf \:P(A |B) = \dfrac{13}{18}}}}}$

Additional Information :-

$\red{\boxed{\sf \:P(A |B) = \dfrac{P(A \: \cap \:B)}{P(B)} = \dfrac{n(A \: \cap \:B)}{n(B)} }}$

$\red{\boxed{\sf \:P(B |A) = \dfrac{P(A \: \cap \:B)}{P(A)} = \dfrac{n(A \: \cap \:B)}{n(A)} }}$

$\red{\boxed{\sf \:P(A \: \cap \:B') = P(A)- P(A \: \cap \:B)}}$

$\red{\boxed{\sf \:P(A' \: \cap \:B) = P(B)- P(A \: \cap \:B)}}$

$\red{\boxed{\sf \:P(A' \: \cap \:B') = 1- P(A \: \cup \:B)}}$

$\red{\boxed{\sf \:P(A' \: \cup \:B') = 1- P(A \: \cap \:B)}}$