I'm studying Introduction to probability and currently, I'm stuck with the following problem. Given:
$P(A)=0.7$,$P(B)=0.5$,$P(A\cap B)=0.45$
What is the probability of A and not B?
I've checked this similar question but I don't understand the answers. Also I've asked my instructor and she told me that $1-P(A\cap B)(P(B^c))$ is the answer (As the answers suggested, this result is not correct). Why is that? She does not provided me a completely explanation.
Update 1: The original problem is the following
In a multiplex cinema, there are two different rooms, $A$ and $B$, working simultaneously. Let $SA$ be the event that, during a certain showing, room $A$ becomes full before the film begins, and let $SB$ be the event that, during the same showing, room $B$ becomes full before the beginning of the movie. We know that $P(SA)=0.7$; $P(SB)=0.5$ and $P(SA∩SB)=0.45$
Calculate the probability that room $A$ will become full and room $B$ will not.
Did I state the problem correctly?
Update 2: Add a Venn diagram. Following the advice of Ethan Bolker. Here is the Venn diagram that I made.
3 Answers
$\begingroup$Hint: try drawing a Venn diagram.
$\endgroup$ $\begingroup$I would imagine A to be a line segment of length 0.7 and B to be a line segment of length 0.5 that overlap by a distance of 0.45.
For example A could be [0, 0.7] and B [0.25, 0.75]. Then A union "not B" is [0, 0.25] so has probability 0.25.
$\endgroup$ 3 $\begingroup$Hint:$$P(A \cap \overline{B}) = P(A\setminus(A\cap B)) = P(A) - P(A\cap B)$$
$\endgroup$ 2
