## Disputed Cards

In one of my classes we’ve plunged into probability, which is a fun subject because suddenly there’s no complicated calculations to do — at worst, there’s long ones — but you have to be extremely careful about what calculations you do, so all that apparent simplicity gets turned into conceptual difficulty. So it’s brought to mind something a student in an earlier term told me about.

I’d given as a problem one of the standard rote probability puzzles: the chance of picking three red cards in a row from a well-shuffled and full deck. The chance of doing this depends a bit on whether you put the just-picked card back in the deck and reshuffle or not, but in either case, it’s pretty close to about one in eight. Multiple students got this exactly right, glad to say, but one spun it out into an anecdote.

The student was pretty enthusiastic about the course topics and while hanging out with a sibling mentioned this as a problem, and the solution. The sibling, however, didn’t believe it, and insisted that since there are an equal number of red and black cards there should be a one in two chance of drawing three red cards in a row out. The two disputed the subject for the whole weekend, and my student apparently rather appreciated having something novel to argue about.

I’m always delighted when a student is interested enough in a problem to mention it to anyone else, and probability puzzles often give things with real-world models simple enough to catch the imagination. But I was (and still am) surprised the question could last a whole weekend. Freaky things can happen in small sample size, but I’d be willing to bet that trying it out with a deck of cards a couple times would at least provide convincing evidence that the chance of three reds in a row wasn’t one in two.

Possibly my student wasn’t communicating the problem well; one in two would be about right for the chance of picking a third red card, after all, regardless of what the first two were. Or perhaps they didn’t have a deck of cards to try it out. I couldn’t reliably lay my hand on a deck of cards until a few weeks ago, when I bought a deck so I could demonstrate problems in class.

Also, a standard-size deck of cards is far too small for a class demonstration. I need to find a magic store and get an oversized deck of cards. I have the same problem with the dice I picked up, but there I should be able to find a giant pair of dice in an auto parts store. They’ll be fuzzy, but should express the idea of dice well enough for that.

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