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Today I faced the following question. Didn't even know how to read it. Where should I start?

Considering that $P(A) = 0.9$, $P(B) = 0.8$ and $P(A \cap B) = 0.75$, compute:

a) $P(A \cup B)$

b) $P(A \cap \overline{B})$

c) $P(\overline{A} \cap \overline{B})$

d) $P(A| \overline{B})$

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    You should use the "notation" tag if that's what you're having trouble with. – 2017-01-06
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    Why downvoting, I'm wondering. – 2017-01-06
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    Probably because you're saying that you don't know how to even read the symbols suggests that a good place to start is your textbook or lecture materials where the definitions are surely given for the notation used. One of the commonly suggested reasons for downvoting is "*This question does not show any research effort*" which is certainly the case here. – 2017-01-06

1 Answers 1

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$A$ and $B$ are events, i.e. subsets of the sample space. As with sets, $\cup,\cap,$ and $\overline{~~}$ are all operations which let us combine or otherwise manipulate sets.

$A\cup B$ is "A union B" is defined as $A\cup B=\{x~:~x\in A~\text{or}~x\in B\}$

$A\cap B$ is "A intersect B" is defined as $A\cap B=\{x~:~x\in A~\text{and}~x\in B\}$

The horizontal bar over a set indicates the complement. $\overline{B}=\{x~:~x\not\in B\}$

$P(E)$ is the probability that event $E$ occurs when you run an experiment, i.e. a measure of how likely the event is to occur which will have a value given to it between 0 and 1 satisfying some nice properties. Similarly, notice that $A\cap B, A\cup B$ also qualify as events, so something like $P(A\cup \overline{B})$ is "the probability of A union the complement of B" is the probability that the event $A$ occurs or the event $B$ doesn't occur.

The vertical line on the other hand represents conditional probability.

$P(A\mid B)$ is read "The probability of $A$ occurring given that $B$ occurs"

To continue, remember a few definitions and key properties:

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

$P(A)+P(\overline{A})=1$

$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$

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    In all honesty: while reading the question, the only thing I struggle with, is the $P(A)=0:9$ part. In my book, this comes down to 0 positives in 9 total outcomes, resulting in prob of 0. (and similar for the rest) Could you elaborate on that? – 2017-01-06
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    @Laray I would expect that it is a typo or an error in rendering. Surely any colon you see should be a decimal point instead. Use instead $P(A)=0.9$ and similarly for the others and continue from there. – 2017-01-06
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    Well that would have been to easy to come by myself. *facepalm* Thanks! – 2017-01-06