Some doubts about conditional probability

Hi everyone, I’m studying the module about conditional probabilities.
Now, it’s clear that to formula to calculate is p(A|B) = p(A ∩B) / p(B)
So, if I understand correctly, p(A|B) is given by calculate the probability of A and B to happen divided by the probability of B.

Now I have the scheme
image

I know that by looking at the table p(M|L) = 32/90 but what if I use the formula?

so p(M ∩L) / p(L).
I know that p(A ∩B) = p(A) * p(B), so p(M ∩L) / p(L) = p(M)*p(L) / p(L) right?

Why do I get two different results? In fact one is 32/90 = 0.35… while the other is 0.2575

Could you please explain to me if I’m missing something?

P(A \cap B) = P(A)*P(B) is true only if A and B are independent events.

Do you think M and L are independent events based on the given information?

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Hmmmm…I would say they are independent, no?

Does the data support that?

You already stated:

If M and L were independent, then P(M|L) = P(M). Because the occurrence of L would not change the probability of M if they were independent.

But that’s not the case, because we can see that P(M) = \frac{515}{2000}. So, they can’t be independent.

It can feel counter-intuitive because you might think “Buying one item doesn’t influence buying a different item”, but that might not necessarily be the case. It’s not difficult to assume that if someone is buying a laptop, they might want to buy a mouse as well. And the data shows that there is an overlap.

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So, let’s see if I understand correctly: we know that p(M) = 515/2000 while p(L) = 90/2000?
Am I correct?

Yes, that’s correct.