# Probability of A and not B

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.

• Are you sure you stated the problem and the solution correctly? For the question you asked you do not need to know the probability of $B$, and the answer you were given is wrong. – Ethan Bolker Mar 4 at 17:18
• Maybe I stated the problem wrong. I update my question to add the original problem. I stated the solution correctly, it is wrong? – Dennis Concepción Martín Mar 4 at 17:25
• I am still confused. Your restatement does not add information. Perhaps there is missing information about customers arriving and choosing one of the rooms with a certain probability. I think you will have to get more clarification from your instructor. – Ethan Bolker Mar 4 at 17:32

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.

• That's the exact same solution my instructor gave me. How can I see this in "probability language" instead of segments? – Dennis Concepción Martín Mar 4 at 17:40
• @DennisConcepciónMartín: In probability language, with the visualization suggested, you should be able to see that the answer is $P(A) - P(A\cap B)$ – true blue anil Mar 4 at 18:22
• After a coffee break I could understand it. Thank you. – Dennis Concepción Martín Mar 4 at 18:52

Hint: try drawing a Venn diagram.

Hint: $$P(A \cap \overline{B}) = P(A\setminus(A\cap B)) = P(A) - P(A\cap B)$$

• It gives me the correct solution but I do not fully understand the reasoning. I'll study more. Thank you. – Dennis Concepción Martín Mar 4 at 17:44
• @DennisConcepciónMartín Actually, there is nothing done with the probabilities, only some transformation of set operations. – VIVID Mar 4 at 17:50