What is the probability that no "set" exists in a group of 9 cards?

At my work, we have a card game called SET that we play with our kids. If you haven't heard of this game, here are the rules; otherwise, read on to the questions: --- Rules: A card game called SET consists of 81 cards that consists of every possible combination of the following features: (a) The cards... show more At my work, we have a card game called SET that we play with our kids. If you haven't heard of this game, here are the rules; otherwise, read on to the questions:

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Rules:
A card game called SET consists of 81 cards that consists of every possible combination of the following features:
(a) The cards either have oval, diamond, or squiggly shape(s) on them
(b) Each of the shape(s) are either red, purple, or green
(c) Each of the shape(s) are filled in either with a open, striped, or solid shading.
(d) There are either one, two, or shapes on a card.

A set of three cards are considered a "set" when either of the four conditions are satisfied:
(i) They all have the same number, or they have three different numbers.
(ii) They all have the same symbol, or they have three different symbols.
(iii) They all have the same shading, or they have three different shadings.
(iv) They all have the same color, or they have three different colors.

For example, a group of cards that consist of a card that has one solid-filled red oval on it, a card that has two solid-filled red ovals on it, and a card that three solid-filled red ovals on it are considered a "set." A group of cards that consist of a card that has a solid-filled red oval on it, a card that has two solid-filled red ovals on it, and a card that consists of three solid-filled PURPLE ovals on it is NOT considered a "set."
(If this explanation is confusing, see:
http://en.wikipedia.org/wiki/Set_%28game... )
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Here is the question:
Find the probability that among 9 random cards, none of the 9 cards produce a "set."

Bonus question:
Prove that at LEAST one "set" exists among 20 or more random cards.
(This is one of the Basic Combinatorics of Set on the Wikipedia article.)
Update: My bad! I meant to say that a "set" obeys all four conditions.
Update 2: You are correct that a "set" may not exist among 20 cards. I misread the statement on the article. I mean 21 or more then.
Update 3: And it is a very good game. At my work, we have a kid that just finished kindergarden that is getting the hang of it. I'm not that good at it, though. I'm not that fast.
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