Probability Multiplication Rule: Logic & Intuition + Bayes Theorem

Hello,
So I realize that this is not necessarily a ‘code’ question; however, I am making my way through the Mastering Stats w/ Python course and am having a hard time forming an intuition as to why we multiply the probabilities of two events to calculate the probability that they occur simultaneously. Why do we not sum the individual probabilities? Furthermore, when would one choose to use the addition rule over the multiplication rule and vice versa?

i.e P(A and B) = P(A) * P(B)

I have looked a little online, but I did not come across anything that intuitive.

By extension, I am also curious about the logic behind Bayes theorem. Perhaps, this will become more clear once the above confusion is explained, but please do free to comment!

<3

You would use addition if you want the probability of one or the other:

P(A or B) = P(A) + P(B)

This is because you want the probability of getting A, then you want the probability of getting B, but as the events are kind of unrelated, you just add them.

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Okay, thanks for clarifying this! So say we want the probability of both events happening simultaneously – how does multiplication capture this?

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I’m not too sure, to be honest, as I just kind of accepted it when I learned it. My intuition says that the probability of getting two events has got to be less than one, and the only simple operation which makes fractions smaller (barring subtraction) is multiplication. But that’s just avoiding the question…

I think of it this way: if P(A) = 0.1 and P(B) = 0.5, then if I do something 10 times, I’d expect to get A once, and if I did something 2 times, I’d expect to get B once. Now, if I want to get A and B (say A is the probability of drawing a red ball from bucket A, and B is the probability of drawing a red ball from bucket B), then the probability that I get A and (then) B will P(A)*P(B). It’s like this: if I get A, there’s only a 0.5 chance that I’ll get B. Conversely, if I get B, there’s only a 0.1 chance that I got A, so each time I get one, I may not get the other. To get both, you have to multiply…

Sorry for the massive block of text (which probably makes no sense), it’s just I’ve never tried to articulate the reason…

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