Our goal is to make $20.20. But … there are twists!

We are given the following denominations: penny ($0.01), nickel ($0.05), dime ($0.10), quarter ($0.25), 1 dollar bill, 5 dollar bill, 10 dollar bill, and a 20 dollar bill.

You are allowed to request as much of any of the denominations you want and whatever mixture of denominations you’d like provided that the total is $20.20.

Here is the first twist: You get each denomination with a certain probability. Here are the probabilities

• Penny — 99.99%
• Nickel — 99.95%
• Dime — 99.90%
• Quarter — 99.75%
• $1 bill — 99.00%
• $5 bill — 95.00%
• $10 bill — 90.00%
• $20 bill — 80.00%

The second twist is that once you’ve decided how you want to receive the $20.20, there is a draw based on the probabilities of you receiving your choice. So, if you request one $20 bill and two dimes, then you are given these based on the probability you’ll get each of the requests. So for the request of [$20, $0.10, $0.10], there’s an 80% chance you’ll get the $20 bill, a 99.90% chance of getting each of the dimes. So the probability that you get $20.20 with this combination of denominations is \(.8\cdot .999 \cdot .999 = .7984008 = 79.84008\%\)

Now there are a few challenges:

Challenge #1

What combination of denominations gives the highest probability of getting exactly $20.20? And what is that probability?

Challenge #2

What combination of denominations gives the highest expected value? And what is that expected value?

Don’t forget to check out these 20 puzzles for 2020!