# Thread: [Probability] Easy problem but I can't figure it out! (drawing cards from a deck)

1. ## [Probability] Easy problem but I can't figure it out! (drawing cards from a deck)

Suppose I have a deck with 40 cards.
3 cards type A,
2 cards type B,
3 cards type C,
and the rest from other types.
I'll take 6 cards from this deck. What's the probability that, at the same time, I'll get at least one card of each types A, B and C?
I know that the hypergeometric distribution should be used. I simulated 5000 times on Excel and reached an aproximated result of 0.032 but I couldn't get to this result analytically.

2. ## Re: [Probability] Easy problem but I can't figure it out! (drawing cards from a deck)

Originally Posted by vasantonio
Suppose I have a deck with 40 cards.
3 cards type A,
2 cards type B,
3 cards type C,
and the rest from other types. I couldn't get to this result analytically.
Notation: Let $\displaystyle X$ be the event of getting at least one of type $\displaystyle X$ and $\displaystyle X^c$ is none of type $\displaystyle X$.

$\displaystyle \mathcal{P}(A\cap B\cap C)=1-\mathcal{P}(A^c\cup B^c\cup C^c)$.