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I noticed in my introductory statistics course that most operators like AND/OR relate to mathematics operators like multiplication and addition. Is there any relation between conditional probability to mathematical operators?

My follow up question for this is: I'm struggling to determine when to define an event as a conditional probability event or when to define it as A and B have occurred. I'm not sure why this is not making sense for me, but it'd help to know a better way to understand when and how to define conditional probability.

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Probability theory is a branch of mathematics, so it is a "mathematical operator". It is defined in terms of joint probability of two events $A$ and $B$ occurring together

$$ P(A\; \mathrm{and} \;B) = P(A, B) $$

and the marginal probability $P(B)$,

$$ P(A | B) = \frac{P(A,B)}{P(B)} $$

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  • $\begingroup$ I meant more of the lines: P(A and B) = P(A) * P(B) P(A|B) = P(A) (operator) P(B) Is there something I can look up to understand conditional probability in this way? $\endgroup$
    – Jeygopi
    Commented Oct 9, 2019 at 12:03
  • $\begingroup$ @Jeygopi more of what? $\endgroup$
    – Tim
    Commented Oct 9, 2019 at 12:04
  • $\begingroup$ More along the lines, is there something analogous to a binary operator that P(A|B) can be related to? $\endgroup$
    – Jeygopi
    Commented Oct 9, 2019 at 12:35
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    $\begingroup$ @Jeygopi the above is the definition, you cannot go simpler then that. $\endgroup$
    – Tim
    Commented Oct 9, 2019 at 12:51
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The simplest way to understand conditional probability at an introductory level is to undertake exercises where you relate probabilities to areas and relative-areas in a Venn diagram. There are many resources available that teach conditional probability in this way, but I will give a simple outline here. Consider a Venn diagram with two events $A$ and $B$ represented as interlocking circles (picture taken from this related question), and suppose we treat the areas as being equivalent to the probabilities. The probability of $A$ is the area of the purple circle and the probability of $B$ is the area of the blue circle. The conditional probability of $A$ given $B$ would be the relative size of the intersection where both these events occur (the dark-blue area in the middle) relative to the area of the blue circle.

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If you do a quick search online you will find hundreds of introductory resources that explain conditional probability by reference to Venn diagrams, and this gives you a nice visual illustration of the concept. You will also be able to find exercises to test your knowledge. I would recommend this as a starting point for anyone having difficulty understanding the relationship between conditional probability and the AND/OR operations on events.

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When you say $$P(A \text{ and }B)=P(A)\cdot P(B)$$ that is true for independent events. For non independent events, the chain rule for probability says: $$P(A\text{ and }B) = P(A|B)\cdot P(B)$$ In fact, when $A$ and $B$ are independent, $P(A|B)=P(A)$ and you get the first definition back.

The event $A|B$ makes sense when the probability of $A$ is influenced by the outcome of $B$. Its probability $P(A|B)$ is defined as Tim said before.

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