I have some ambiguity in the following problem

Three cards are drawn in succession from a deck without replacement. find the probability distribution for the number of spades.

I have come with this solution.

Let S1: appearance of spade on first draw S2: appearance of spade on 2nd draw S3: appearance of spade on 3rd draw

S01: No spade on 1st draw S02: No spade on 2nd draw

P(S1)=13/52

P(S2)=P(S2/S1)+P(S2/S0)=12/51+13/51

P(S3)=P(S3/S01*S02)+P(S3/S1*S02)+P(S3/S1*S2)=13/50+12/50+11/50

Need your opinion Please!