finding the optimal strategy You have a deck of 32 playing cards. Somebody draws one card after another and shows them to you. At any point of time you may bet that the next card is black. If it is indeed black you earn $10, otherwise nothing. If you don't do anything you earn nothing as well. Find the optimal strategy. In other words you should find the point of time where the quota of remaining red cards in the deck is maximal.

Julius Blankenship

Julius Blankenship

Answered question

2022-09-04

Finding the optimal strategy
You have a deck of 32 playing cards. Somebody draws one card after another and shows them to you. At any point of time you may bet that the next card is black. If it is indeed black you earn $10, otherwise nothing. If you don't do anything you earn nothing as well. Find the optimal strategy.
In other words you should find the point of time where the quota of remaining red cards in the deck is maximal.

Answer & Explanation

Caiden Li

Caiden Li

Beginner2022-09-05Added 17 answers

I take as additionnal assumption that if you do nothing, you bet on the last card rather than automatically losing.
There is no optimal strategy. Imagine the game this way. Instead of betting on the next card, you bet on the last card. Note that you still interrupt the game at any time you want you just bet on a different card.
The last card has the same probability of being black as the first one on the remaining card since you know nothing of how they are ordered. (i.e. every permutations is possible). The expected number of times you win this game is thus the same as in yours.
When do you win my game? Well you win if your last card is black, which ultimately happens half of the time.
This simple (symmetry) argument shows that the game is fair and there is no optimal stategy

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