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Assume we have a black and a red cube with 6 sides. We definite two events A = "the black dice shows 5", B = "The product of the number of pips is a prime number". We roll the dice. So $P[A] = \frac{1}{6}$ and $P[B] = \frac{1}{6}$ right ? Now i want to check, if A and B are independent. So $P[A \cap B] = \frac{1}{36} = \frac{1}{6} * \frac{1}{6} = P[A]P[B]$ so A and B are independent. My instincts tell me the events are dependent. Where is my mistake ?

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    $\begingroup$ Looks right. Hard to have intuition about things like this. $\endgroup$ – lulu Mar 23 at 14:01
  • $\begingroup$ What do you mean with "the number of pips" ? $\endgroup$ – Peter Mar 23 at 14:03
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    $\begingroup$ "My instincts tell me the events are dependent. Where is my mistake?" Your mistake is in being too trusting of your instincts. This is a fine example of a pair of independent events whose independence may be counterintuitive. $\endgroup$ – JMoravitz Mar 23 at 14:03
  • $\begingroup$ @Peter a six sided die is commonly made with a number of small colored circular indentations called "pips." The numerical result of throwing a die is the number of visible pips on the upward showing face. Worded another way, it is the numerical result that the die shows. $\endgroup$ – JMoravitz Mar 23 at 14:04
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    $\begingroup$ "The product of the number of pips is a prime number" is the event $\{(1,2),(1,3),(1,5),(2,1),(3,1),(5,1)\}$ $\endgroup$ – Graham Kemp Mar 23 at 14:08
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$B$, "The product of the number of pips is a prime number", is the event $\{(1,2),(1,3),(1,5),(2,1),(3,1),{\bf(5,1)}\}$

So $\mathsf P(A\mid B)$ is clearly $1/6$.

The weighted ratio of outcomes in $A\cap B$ to $B$ equals the weighted ratio of outcomes in $A$ to $\Omega$ (the outcome set).

That is all that is required for independence.

The notion that "independent events don't influence each other" is not too misleading.   It is just that our judgement on whether this is the case is not very reliable.   Our instincts can be way off.

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