# Find Mistake: Independence of two Events

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 ?

• Looks right. Hard to have intuition about things like this. – lulu Mar 23 at 14:01
• What do you mean with "the number of pips" ? – Peter Mar 23 at 14:03
• "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. – JMoravitz Mar 23 at 14:03
• @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. – JMoravitz Mar 23 at 14:04
• "The product of the number of pips is a prime number" is the event $\{(1,2),(1,3),(1,5),(2,1),(3,1),(5,1)\}$ – Graham Kemp Mar 23 at 14:08

$$B$$, "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).