It is given that the events A and B are such that [P( A ) = 1 4,P( A|B ) = 1 2] and [P( B|…

Question

It is given that the events $A$ and $B$ are such that

$$P\left( A \right) = {1 \over 4},P\left( {A|B} \right) = {1 \over 2}$$ and $$P\left( {B|A} \right) = {2 \over 3}.$$ Then $P(B)$ is :

Accepted Answer

Given that,

$$P\left( {{A \over B}} \right) = {1 \over 2}$$

$$ \Rightarrow $$ $${{P\left( {A \cap B} \right)} \over {P\left( B \right)}}$$ = $${1 \over 2}$$.............. equation (1)

$$P\left( {{B \over A}} \right) = {2 \over 3}$$

$$ \Rightarrow $$ $${{P\left( {A \cap B} \right)} \over {P\left( A \right)}}$$ = $${2 \over 3}$$.............. equation (2)

Dividing equation (1) by equation (2) we get,

$${{P\left( A \right)} \over {P\left( B \right)}}$$ = $${3 \over 4}$$

$$ \Rightarrow $$ $${P\left( B \right)}$$ = $${4 \over 3}$$ $$ \times $$ $${P\left( A \right)}$$

= $${4 \over 3}$$ $$ \times $$ $${1 \over 4}$$

= $${1 \over 3}$$

$$\therefore$$ Option (B) is correct.

Solutions · 1