Describe the induced probability $P_X(D)$ on the space $\mathbb{D}=\{0,1,2,3,4\}$ of the random variable $X$.

Question

I know that $P_X(D)$ is determined if I find $P_X(\{d_i\})$ for each $d_i\in\{0,1,2,3,4\}$. So $P_X(\{0\})=P[X^{-1}(\{0\})]=\{\text{ the cards that are neither ace nor king nor queen nor jack} \}$, but for example $P_X(\{0\})$ is difficult to describe, is there an easier way to do this? Thank you very much.

Answer 1

$P_X(4)=P\{\mathrm A\heartsuit, \mathrm A\diamondsuit, \mathrm A\clubsuit, \mathrm A\spadesuit \}\\P_X(3)=P\{\mathrm K\heartsuit, \mathrm K\diamondsuit, \mathrm K\clubsuit, \mathrm K\spadesuit\}\\P_X(2)=P\{\mathrm Q\heartsuit, \mathrm Q\diamondsuit, \mathrm Q\clubsuit, \mathrm Q\spadesuit\}\\P_X(1)=P\{\mathrm J\heartsuit, \mathrm J\diamondsuit, \mathrm J\clubsuit, \mathrm J\spadesuit\}\\P_X(0) ~{= P\{2\heartsuit, 2\diamondsuit, 2\clubsuit, 2\spadesuit, 3\heartsuit, \ldots ,10\spadesuit\}\\=1-P_X(4)-P_X(3)-P_X(2)-P_X(1)}$