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P(A) = P(The world would be destroyed today by a meteor) = 1/2

P(B) = P(My pet plant would sprout tomorrow) = 1/2

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When combined, there are 3 possible cases.

  1. The world will be destroyed, and my plant will die too = 2/4
  2. The world will not be destroyed, but my plant still won't sprout = 1/4
  3. The world will not be destroyed, and my plant will sprout = 1/4

The point I am confused is the part where you compare P(A∩B) and P($A^c$∩B).

So like, P(A∩B) = 0 ≠ P(A)×P(B), which means A and B are not independent.

But, P($A^c$∩B) = 1/4 = P($A^c$)×P(B), which means A and B are independent.

What am I missing here?

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The tricky part is that your probability for $B$ is not a straight-up probability, it's a conditional probability. It's conditioned on the fact that $A$ does not occur, so really, your probability for $P(B)$ is $P(B|A^C)=0.5$.

We can see that if you fill out your diagram. $P(A\cap B)=0$, so the top left should be $0$. $P(A\cap B^C)=0.5$, so the bottom left is $0.5$. The right hand column should have $0.25$ in each box. Therefore, when you add up the probabilities in the top row (to get the probability $B$), you get $0.25$, which means $P(A^C\cap B)=0.25\neq 0.5\cdot 0.25$.

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You stated that

P(B) = P(My pet plant would sprout tomorrow) = 1/2

That is not P(B), instead, that is P(B|A$^c$) as you have made the assumption that the earth was not destroyed.

If you simulated these set of events many times, you would find P(B) = 1/4.

This would show dependence as:

P(A$^c$) x P(B) = 1/2 x 1/4 = 1/8 and hence,

P(A$^c$∩B) = 1/4 ≠ P(A) × P(B)

The product rule for dependent events is P(A$^c$∩B) = P(A$^c$) x P(B|A$^c$). This is the formula that you used without realisation as you confused P(B) and P(B|A$^c$). That is why you thought that formula showed the probabilities were independent.

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