Probability question - Cards

Hello everyone!
I just started to study statistics by myself because I spent a semester abroad and there are no more lectures at the University, if you have any hints about how to study statistics you are more than welcome to share them with me.

Here is the exercise:
We choose at random and without replacement 5 cards from a deck of italian playing cards (40 cards with four suits: coins, swords, cups and clubs, 10 cards for each suit).
Compute the probability of A: no coins. Describe in words the complement event of A and derive the corresponding probability.

I tried to solve the exercise drawing a a probability tree diagram (is there any faster solution?).

Doubts (I don't know whether my solutions are right or wrong):
1. P(A)=0.43
Is it correct?
2. Complement event of A. I know that a complement of an event are all the outcomes that are NOT in the event.
Is the complement of event A getting only coins or at least one coin?
3. P(Ac)=0.07%
Is it correct?

Thank you for your attention, I hope someone will guide me through the correct resolution.