You have a State Lotto machine into which N ping pong balls have been placed.
Each ball is numbered with one of the first N primes. Each time you press the button, one more randomly chosen ball appears in the output tray.
Your score is the
mean of the numbers displayed on the balls chosen so far.
Your goal is to maximize your score by choosing when to stop pressing the button.
What is the optimal strategy?
For your strategy, what is the ratio of the expected value of your score to the average of the first N primes?
μ=the mean of the N primes. I assumed we are drawing without replacment.
The primes are skewed positively, especially as N increases. This means there are at least as many numbers in the set below μ as above it.
As a consequence, if you current mean is ever above μ you should stop. (The expected value of another draw will never be above μ and so your mean will never be expected in increase.)
If your current mean is below μ you should always keep drawing. The worst you can do is to pick all N balls and end up μ.
For some small N I worked out the values by hand
N=2 μ=5/2 expected max = 11/4 difference = 1/4 ratio = 11/10
N=3 μ=10/3 expected = 145/36 difference = 25/36 ratio = 29/24
N=4 μ=17/4 expected = 811/144 difference = 199/144 ratio = 811/612
(strategy ends up being to stop if you ever get the 7 or if you get the 5 on your first pull.)
I don't see any patterns that would make me want to go any further.
There are some potential sequences here but even with 3 terms I got no hits on OEIS.
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Posted by Jer
on 2024-04-03 13:45:09 |
Thoughts and some small N
