Question 1: we have a process that succeeds 95% of the time, and all experiments are independent. We can set one of two challenges:

A: Try to get 7 or more successes out of 9 trials;

B: Try to get 3 or more successes out of 4 trials.

We might guess A is a harder bar to reach, since 7/9 = 77.8% and 3/4 = 75%. But, perhaps surprisingly, Pr(A) > Pr(B)! (In fact, A has a 99.2% chance of success, while B has a 98.6% chance of success.)

But now assume everything is exactly the same, but the original success rate is 80%. Now Pr(B) > Pr(A). What is up with that?

Question 2: We have a process with an 80% success rate, and we need to design a demo that gets (at least) 75% successes. The Law of Large Numbers says that if we let the number of trials go to infinity, we're guaranteed that the demo will work.

But this is a demo, and I don't have an infinite amount of time. Let's say I have at most time for 10 trials, so I can pick n trials where 1 <= n <= 10. To maximize the probability of observing at least 75% successes, what should I choose n to be? If we trust the intuition of the LLN, then the best answer should be n=10, right?

Turns out the answer is n=4. In fact, n=10 is the sixth best choice of ten, not even in the top half. The ranked choices are n=4,1,8,9,5,10,6,2,7,3. (The fact that n=1 is the second best choice here is amazing: n=1 is just a complete gamble.)

It turns out these two questions are in some sense the same. The driver for the weirdness is embedded in the Law of Large Numbers itself --- it tells us that certain things converge, but it doesn't guarantee that the convergence is monotone. In fact, the convergence has a certain periodicity that will be cool to explore.

**Faculty Member: **Lee DeVille

**Difficulty: **Beginner

**Team Meetings: **weekly

**Prerequisites: **A probability course *or* a combinatorics course *or* a discrete math course. There will definitely be some programming/simulation in this project, but fluency in any language will be fine.

More information on Monotonicity vs periodicity in the Law of Large Numbers