In a city, the official lottery consists of scratch cards that cost 10 bucks each. Each scratch card contains 36 squares with a hidden number on each square : 21 of them are “zeroes”, 9 of them are “ones” and 6 of them are “tens”. To proceed, one can scratch as many squares as one wants. The total revenue consists of the product of all the scratched numbers. For example, if someone scratches 2 “ones” and 1 “ten”, the total revenue would be 10 bucks. If everyone in the city plays with the optimal strategy, what percentage of the total sum that the lottery company would make would be given back to the players? Please don’t use computers.
This problem is adapted from a statistics exam for AP Statistics.