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I'm creating an activity for my class that teaches the vertical and horizontal line tests and for the life of me I can't figure out the probabilities involved. Here's the exercise:

Draw 5 pairs of cards from a standard deck. Each pair defines a point in the plane: the first card is the x-coordinate of a point and the second card is the y-coordinate (Jacks = 11, Queens = 12, Kings = 13). Red cards are positive and black cards are negative. Plot the 5 points, shuffle the cards, repeat the process to get a total of 10 points.

Question: What are the odds your graph passes the vertical line test?

This isn't the question the students will be answering, they are just calculating the experimental probabilities by polling the class. I thought it would be cool to show them the theoretical odds so we could discuss how close we are, but so far I've only been able to find simple, easy-to-understand wrong answers. I imagine it's pretty likely I've just forgotten some combinatorics and this is actually pretty easy to figure out.

(Update: About 34% is the most sensible answer I've gotten so far.)

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    Why do you have red for positive and black for negative? That's opposite of the usual metaphors about bottom lines (based on obsolete mechanical calculators to be sure).2012-09-16
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    @Henning I think the practice of recording negative values in red is significantly older than the widespread use of mechanical calculators. I can find clear references to the practice in Google Book search results well back into the middle of the 19th century.2012-09-17
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    @Henning It was an arbitrary choice, but I went with the labeling of car battery terminals. The red one is positive and the black is negative.2012-09-25

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