In the last post I described a problem with Dan Meyer’s otherwise excellent expected-values teaching tool: you’d like to wow the kids by correctly predicting the answer a month in advance, but the given setup is actually too variable to let you make a safe prediction.
Essentially, if you’re saying “Let’s do a magic trick to get kids engaged in this topic,” but the trick takes a month to run AND only has a 30% chance of working… then why not tweak the trick to be more reliable?
spin it many more times?
Part of this unreliability comes from the low number of spins — about 20 spins total, if you do it once every weekday for a month. The “easy” fix is to spin Dan’s spinner many more times … but it turns out you’d have to spin it about 4000 times to be 90% confident your prediction will be right. Even if you have the patience, that’s a lot of individual spins to track in your spreadsheet or wherever.
use a MORE reliable spinner?
Another fix might be to change the spinner so that it works reliably given only 20 spins. First, we don’t want any of the sectors too small, else we might not hit them at all during our 20 spins, and then it becomes unpredictable. It turns out the smallest sector has to be at least about 1/9th of the spinner if you want to be 90% confident of hitting that sector at least once in those 20 spins.
(Let . Then $latex p(y<0) = 1-p(y=0) = 0.905$.) If we round that up to 1/8th instead, we can easily use a Twister spinner (which has 16 equal sectors). After playing with some different options, using the same simulation approach as the previous post, I found that the following spinner seems to work decently: 1/2 chance of $100, 3/8 chance of $150, and 1/8 chance of $1500. After 20 spins of this spinner, there's about a 87% chance that the "$1500" will have been the winning bet, so you can be pretty confident about making the right bet a month in advance.
Unfortunately, predicting that spinner correctly is kind of unimpressive. The “$1500” is a fairly big slice, so it doesn’t look too risky.
spin just a few more times and use a safer spinner!
What if we spin it just a few more than 20 times — say 60 times, so two or three times each day? That’s not too much data to keep track of. Will that let us shrink the smallest slice, while keeping predictability high, and thus making this all more impressive?
Turns out that if we know we’ll have about 60 spins, we can make the smallest slice 1/25th of the spinner and still be confident we’ll hit it at least once. Cool. If we want to keep the Twister board, and have the smallest slice be 1/16th of the circle, we actually have a 90% chance of hitting it at least twice. So that makes things even more predictable (for the teacher), while still making it less predictable (to the kids) than the previous spinner.
More messing around led to this suggested spinner: 1/2 chance of $100, 5/16 chance of $200, 1/8 chance of $400, and 1/16 chance of $2500. The chance that “$2500” is the right bet after 60 spins of this spinner is about 88%, so again you can make your bet confidently — but this time, the “right” answer doesn’t look as obvious.
In short, I’d recommend using this spinner for around 60 spins, rather than Dan’s spinner for 20 spins. It’s not guaranteed to be “optimal” but it’s far more reliable than the original suggestion.
If anyone tries it, I’d be curious to hear how it went!