Question

Suppose A and B are two events. Event B has occured and it is known that $P\left(B\right)<1$. What is $P\left(A|B^c\right)$ equal to?

• A. $\frac{P\left(A\right)-P\left(B\right)}{1-P\left(B\right)}$
• B. $\frac{P\left(A\right)-P(A\cap B)}{1-P\left(B\right)}$
• C. $\frac{P\left(A\right)+P\left(B^c\right)}{1-P\left(B\right)}$
• D. None of the above

Answer 1

The correct option is B $\frac{P\left(A\right)-P(A\cap B)}{1-P\left(B\right)}$

Given that $A$ and $B$ are two events and event $B$ has occured,

From the division rule of probability we know that $P\left(\frac{A}{B}\right)=\frac{P(A\cap B)}{P\left(B\right)}$,

similarly,

$P\left(\frac{A}{B^c}\right)=\frac{P(A\cap B^c)}{P\left(B^c\right)}$

$⟹\frac{P\left(A\right)-P(A\cap B)}{1-P\left(B\right)}$