Teaching Probability

[barking_iguana]

It occurs to me that when teaching combinations, it's a bad idea to present the concept as selecting some and not others from a group. rather, it's partitioning a group into two piles. And having the number of new piles equal to two should not be presented as essential to the concept, either. It's only the first example. The number of ways a group of, say, 13 cards can be split into piles of 5, 4, 3, and 1 is 13!/5!/4!/3?/1!.

I think this generalization is at least as easy to comprehend as the more limited case of two piles where only one of the piles is of interest. And it provides a much more versatile tool. But I've never seen a high school level book that teaches the more flexible and powerful tool.

Comments

[chemoelectric.livejournal.com]

Line the items up first; then it is subdivision, which I can visualize as a programmer, even though I suck at visualizing ball draws from an urn. But I think it’s probably better still to do it one partition at a time.

[chemoelectric.livejournal.com]

The subdivision method also avoids confusing combinations with ‘chance’; all that’s required is arbitrariness. (And for probabilities all that is required is lack of a reason to prefer some relative weights.)

[barking-iguana.livejournal.com]

And of course, permutations are just a simple special case. Pick a gold medalist, sliver medalist, and bronze medalist from 7 runners. There's one of each kind of metal and four who don't win. 7!/1!/1!/1!/4!. Even the C student will quickly learn to subtract how many ones there are from seven and then throw out the ones.