Roll a four-sided die with numbers $\{1,2,3,4\}$ each with equal probability. Let $A = \{1,2\}$ and $B = \{1,3\}$. Verify that they are independent.
In my opinion, $P(A) = \frac 2 4$ and $P(B) = P(A)$ because the numbers get rolled at random with equal probability. Then $P(A \cap B) = \frac 3 4$. In my book it says that $P(A \cap B) = \frac 1 4$. Can someone explain to me why I am wrong (if I am wrong)?
$$P(A\cap B)=P(\{1\})=\frac 14$$ Remark: It is perhaps surprising at first sight is that the knowledge that either $1$ or $2$ occurred does not make the event $\{1,3\}$ more likely, because it feels that we "rule out" $4$ altogether. But that is in fact a fallacious way of thinking. If you know that $\{1,2\}$ occurred, then $\{1,3\}$ occurs in case you got $1$, and did not occur in case you got $2$. The probability of $\{1,3\}$ is still $0.5$, the same as it is with no information.