Prove that events $A_p$ are independent.

Question

Let be $n \in \mathbb{N}$. $\mathbb{P}$ is probability measure on $\Omega = \{1,2,\dots, n\}$ on $\sigma$-algebra, where all numbers have the same probability, $\mathbb{P}(\{ k \}) = \frac{1}{n}$ for all $k = 1,2,\dots,n$. Let $\mathbf{P}$ is set of prime numbers. $\phi(n)$ is number of numbers, which are foreign $n$. Define events: $A_p := \{ p, 2p, 3p, \dots \} \cap \{ 1,2,\dots,n \}$.

Prove that events $A_p$ are independent. Thank you for any help.

Answer 1

It does not seem to me that the conclusion of your question is correct. Take $\Omega=\{1,2,3\}$. Then $A_2=\{2\}$, $A_3=\{3\}$, $A_2\cap A_3=\emptyset$ and: $$0=P(A_2\cap A_3)\ne P(A_2)P(A_3)=\frac{1}{3}\cdot \frac{1}{3}$$