- Events A and B are independent. Suppose event A occurs with probability $0.41$ and event B occurs with probability $ 0.40$.
a. If event A or event B occurs, what is the probability that A occurs?
b. If event B occurs, what is the probability that A does not occur?
Round your answers to at least two decimal places.
My try a. $$\begin{split}P(A\mid A\cup B) &=P(A\cap (A\cup B))/P(A\cup B)\\ &=P(A)/P(A\cup B)\\ & =0.41/(0.41+0.40-(0.41*0.40))\\& =0.42\text{ (Correct to 2 decimal place)}\end{split}$$
b. $$\begin{split}P( A^\complement \mid B) &=P(A^\complement)\\ &=1-0.42\\ &=0.58\end{split}$$ Is my answer correct?
Well, for a: the logic is okay, but the calculations went awry. $$0.41/(0.41+0.40-0.41\cdot0.40)~{=0.41/0.646\\ \approx 0.63}$$
And for b: you used $0.42$ instead of $0.41$.