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My nephew thought of the following problem over the holidays:

  • I have 100 coins, half these coins are currently face up heads, the other half are currently face up tails. Currently, my score is 0. Each coin has a 0.5 probability of landing on heads and a 0.5 probability of landing on tails

  • Round 1: I randomly generate an integer "n" between 0 and 100, select "n" coins and flip them - naturally, some change sides, some don't. the coins that weren't selected stay as they are.

  • After Round 1 - each coins that is now facing the opposite side gives me 1 point. And each coin that is facing the same way takes away 1 point. Record the total number of points.

  • Round 2: I now again generate another integer "n" and flip these "n" coins. For coins that were not previously selected but were now selected if they land on the other side, I get 1 point but if they land on the same side, I lose 1 point. And all coins that were not selected in Round 2 automatically deduct 1 point each. Record total points from Round 2 and add to total points from Round 1.

  • Repeat for many rounds

Now, my nephew's game lead me to the following questions:

  • After Round 2, what is my expected score?
  • How many Rounds do I need to play to get some expected score of "x"?
  • On average, what is the optimal number of rounds I should play to give me the highest possible score (before my score starts to decrease and decrease)?

My feeling is that initially it it makes sense to play this game for a few rounds regardless - but then, based on your score, there is probably an probabilistic optimal number of rounds to stop playing this game.

I think that maybe you can use a Markov Chain in where states are either the "score of the game" (e.g. probability of transitioning to -5 points given you are at -1 point) and then estimate the "time to absorption" for different questions based on simulations or solving this analytically - but I am not sure.

Any ideas?

SiuMai2k22
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  • I'm confused by the description of the game. Do you freely choose a new set of $n$ coins every round? And every coin not selected (of which there are $100-n$) is an automatic -1? It seems difficult to get a positive score. – Karl Dec 27 '22 at 05:37
  • I have some clarification questions. a) It sounds like you decide whether to play another turn, then $n$ is generated without your control, then you choose which $n$ coins to flip. Is that right? b) In your "Round 2:" bullet point, you don't say what happens with coins that were selected in round 1 and then also selected in round 2. How do those coins affect the score? c) In general, can you add more explanation about how e.g. round 5 depends on the previous 4 rounds? – David Clyde Dec 27 '22 at 09:33

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