TITLE: What is the probability of drawing at least one of each of these cards?
QUESTION [1 upvotes]: Say you have a deck of cards that contains $n$ total cards.
$x$ amount of these are Card A
$y$ amount of these are Card B
$z$ amount of these are Card C
By drawing $m$ cards at random from this deck, what is the probability of drawing at least one of each of these cards?

REPLY [2 votes]: Let $A$ denote the event that at least one of $A$ is drawn.
Let $B$ denote the event that at least one of $B$ is drawn.
Let $C$ denote the event that at least one of $C$ is drawn.
To be found is $P(A\cap B\cap C)$ and we have the equality:$$P(A\cap B\cap C)=1-P(A^c\cup B^c\cup C^c)$$
To find $P(A^c\cup B^c\cup C^c)$ make use of inclusion/exclusion.
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