[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.

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

Quote:

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 $\displaystyle X$.

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