Totally love the task design – way more structure than my ill-formed ‘let them play with it’.

I see you trying to provoke intellectual need for the idea of expected value when you ask kids to pick out what a fair price is for each of the distributions of duck money.

Here’s the big thing I wonder here: is there actually a way to get kids to develop the ‘formula’ for expected value??

I think there is, and this would be in your task right after you have kids pick out that initial fair value.
Of course, I would have the probability distributions shown like you did, but in words:
– Give ’em a really easy one – every money duck has a $1 bill. Ask them what it’s worth ($1)
– Give ’em a slightly harder one – half of the ducks have $1 bills, half have $5. Ask them what it’s worth ($3)
– A little harder – 75% of the ducks contain $1 and 25% contain 5. Ask ’em if it’s more or less than $3 (less) and have them identify what it’s worth (some might get the calculation of $2)
– Now with three – 1/3 of the ducks contain $1, 1/3 contain $5 and 1/3 contain $10. They should be able to find the average value ($5.33)
– Three, a little harder – 1/2 contain $1, 1/4 contain $5 and 1/4 contain $10; this could be the point where they clamor for a formula or they develop the technique on their own

Then, summarize with the formula EV=p1v1+p2v2+…pnvn and then let them practice.