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Probability Question in Games

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I created this DOS Based game where the user inputs one of the numbers between 1 and 3 inclusive. 1, 2, 3 denote Ace, King, Queen respectively. There are initially 3 of each. The user continues to pull cards until there is no more than one kind. The score is added upon by the odds of being a correct bet. (Ace, King, Queen) Example: The Distribution is (3,3,2) The user bets on Ace which indeed gets pulled. The score is added upon by (3+2)/(3) I am interested in the maximum score that can be achieved (which both relies on the cards the computer draws and the user bets on) I will assume the player bets on the correct card in all cases, so I, heuristically, thought that the maximum would be if the drawn card was the one with lowest probability So the sequence would be: (2,3,3) Ace Cumulative Sum = 2 (1,3,3) Ace 5 (0,3,3) Ace 11 (0,2,3) King 12 (0,1,3) King 14 (0,0,3) King 17 End But when I ran a random card better ( about 10,000 iterations there were some 5 to 10 peculiar situations where it score more than 20!) Any Idea?

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I might be wrong here, but I think you've made a mistake in how you calculate the probability of a card being picked.

Using your example of picking an Ace from the (3,3,2) distribution, you said that the probability of a card being an Ace is 3/(3+2) or 3 out of 5. But that's wrong. There are 3 Aces available and there are a total of 8 (3+3+2) cards in the pack. Thus the probability of a card being an Ace is 3/8. So your scoring algorithm is flawed.

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