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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. enter image description here

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  • $\begingroup$ 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. $\endgroup$ – Ethan Bolker Mar 4 at 17:18
  • $\begingroup$ Maybe I stated the problem wrong. I update my question to add the original problem. I stated the solution correctly, it is wrong? $\endgroup$ – Dennis Concepción Martín Mar 4 at 17:25
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    $\begingroup$ 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. $\endgroup$ – Ethan Bolker Mar 4 at 17:32
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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.

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

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

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

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