Let A and B two events such that [P( A B ) = 1 6,] [P( A B ) = 1 4] and [P( A ) = 1 4,] wh…

Question

Let $A$ and $B$ two events such that $$P\left( {\overline {A \cup B} } \right) = {1 \over 6},$$ $$P\left( {A \cap B} \right) = {1 \over 4}$$ and $$P\left( {\overline A } \right) = {1 \over 4},$$  where $${\overline A }$$ stands for complement of event $A$. Then events $A$ and $B$ are :

Accepted Answer

Given that,

$$P(\overline {A \cup B} ) = {1 \over 6}$$, $$P(A \cap B) = {1 \over 4}$$, $$P(\overline A ) = {1 \over 4}$$

$$\because$$ $$P(\overline {A \cup B} ) = {1 \over 6}$$

$$ \Rightarrow 1 - P(A \cup B) = {1 \over 6}$$

$$ \Rightarrow 1 - P(A) - P(B) + P(A \cap B) = {1 \over 6}$$

$$ \Rightarrow P(\overline A ) - P(B) + {1 \over 4} = {1 \over 6}$$

$$ \Rightarrow P(B) = {1 \over 4} + {1 \over 4} - {1 \over 6}$$

$$ \Rightarrow P(B) = {1 \over 3}$$ and $$P(A) = {3 \over 4}$$

Clearly, $$P(A \cap B) = P(A)P(B)$$,

so, the events A and B are independent events but not equally likely.

Solutions · 1