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Card game (Posted on 2006-08-12) Difficulty: 3 of 5
Jack has challenged you to play a card game with him. The idea of the game is that a player gets a random card and after seeing it it's placed back in the deck. The player may get as many cards as he/she wants and the sum of the values of these cards represents the points this player got. However if the player gets an ace he/she gets zero points as total and the other player may try. How should one play this game (how many cards should be picked) for maximum chance of winning against Jack?
Face cards can be interpreted so that king is 13 points, queen is 12 points and jack is 11 points.

No Solution Yet Submitted by atheron    
Rating: 4.0000 (2 votes)

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Different Approach | Comment 3 of 6 |
(In reply to SLAP JACK!!!! by Dej Mar)

First of all, Jack has an enourmous advantage by going second.

Be that as it may, I think the key is to find the marginal point level where the expected improvement in point total (7.5 if not an ace) decreases the likelihood of Jack being able to best yours by 1/13.

So, take the point value 45 say. We calculate the probability that Jack can best 45 (surely Jack will not be satisfied with a tie) then compare it with the probability of Jack being able to best 52.5, and then compare the difference. So if the difference is less than 1/13 we do not take another card whereas if it is greater than 1/13 we do. It is irrelavent how many cards we draw.

Even more accurate would be to compute the sum of each of the point values between 2 and 13's effects on Jack's success likelihood. I'm not sure if this will give a different result or not.

  Posted by Eric on 2006-08-12 16:25:48

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