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Let's NOT Leave it to Chance (Weekly Brain Potion)

 
 
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Let's NOT Leave it to Chance (Weekly Brain Potion)
LensmireJohn 发表于 2018年10月26日 Friday 16:37
 

In one of our Brain Potions from last year (found here), we explored a betting strategy to supposedly win "Infinite Money" playing a version of Roulette.

In this week's Brain Potion we present a simple card game and explore how much money we can win!

Start with a deck of $2N$ cards, with $N$ black cards and $N$ red cards and start with $\$1.00$. The dealer of the game flips the cards over, one by one, until all $2N$ cards are dealt. Before he flips over each card, you have a chance to bet/predict the color of the next card. You can choose to risk some/all/none of your money, and if you are correct you win double what you bet.

Example: if you have $\$\dfrac{2}{3}$ and choose to bet $\$\dfrac{1}{6}$, then if you are wrong you end up with $\dfrac{2}{3} - \dfrac{1}{6} = \$\dfrac{1}{2}$ while if you are correct you end up with $\dfrac{2}{3} - \dfrac{1}{6} + \dfrac{2}{6} = \$\dfrac{5}{6}$.

Your goal is to come up with a strategy to guarantee you win as much money as possible, leaving nothing to chance.

Example: if $N=1$ we have one red $R$ and one black $B$ card. We can guarantee we finish with $\$2$ using the following strategy. Bet nothing before the first card is flipped. In this way you learn the result of the first card and can bet your full $\$1$ on the second card (and are guaranteed to be correct!). This gives you $1 + 1 = \$2$ at the end.

Try to come up with a strategy for $N = 2$. That means you have to have a strategy that deals will all possible ways the cards could be dealt: $$BBRR, BRBR, BRRB, RBBR, RBRB, RRBB$$ so that you always end up with the same amount of money. Hint: You can guarantee you win more than $\$2$!

As a challenge, once you've solved the problem for $N=2$, try $N=3$, etc. There is a nice pattern for how much money you can win, depending on $N$. Try to find (and prove) it!

Please click here to view and participate in this week's challenge! Good luck!

Have your own request, idea, or feedback for the Brain Potion series? Share with us in our Request and Idea Thread available here.