#Verification of Proof: Optimal Strategy for a Card Drawing Game

55 messages · Page 1 of 1 (latest)

last halo
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I'm working on the following probability problem and would like verification of my proof:

Problem Statement:

Consider a deck of 52 regular playing cards, 26 red and 26 black, randomly shuffled. The cards are face down. Cards are drawn one at a time from the top of the deck and turned face up. At any point before the deck is exhausted, you must declare that you want the next card. If that next card is black, you win; otherwise, you lose.

Question: Is there a strategy that gives a probability of winning strictly greater than 1/2?

My claim is there is no such strategy and below is my proof attempt:

left nebulaBOT
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last halo
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Proof by Induction:
Let P(n) denote the statement: "No strategy exists which gives a winning probability strictly greater than 1/2 for a deck of n cards, where n is even and there are n/2 black cards and n/2 red cards."
We will prove that P(n) holds for all even n ≥ 2, using mathematical induction.

Base Case (n=2):
When n=2, the deck consists of 1 black card and 1 red card. Since the cards are randomly shuffled, and there is no way to distinguish between the two cards, the probability of correctly guessing when the black card will appear is exactly 1/2. No strategy can increase this probability because both cards are equally likely to appear next. Therefore, P(2) holds.

Inductive Step:

Assume that P(n) holds for some even n≥2, i.e., for a deck of n cards (with n​/2 black cards and n/2 red cards), no strategy can give a winning probability strictly greater than 1/2. We need to prove that P(n+2) holds.
By the inductive hypothesis, for the deck of n cards (with n/2​ black and n/2n​ red cards), any strategy has a winning probability of at most 1/2. Adding one black and one red card does not change the relative proportion of black to red cards in the deck. Since the ratio remains balanced and unaffected by the addition of one black and one red card, no additional information or advantage is introduced. Therefore the probability remains same for n + 2.

Thus, by the principle of mathematical induction, P(n) holds for all even n≥2 and therefore we can conclude no strategy exists which gives probability of wining higher than 1/2 for deck of 52 cards.

hollow mango
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you lose with probability 1/2 on the first turn

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before any cards are turned

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you win or lose on the first draw

last halo
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@hollow mango i could not understand.

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is my proof not correct?

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And thanks for you response @hollow mango 🙂

hollow mango
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you are not generating truth here

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slow knotBOT
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@last halo has given 1 rep to @hollow mango

hollow mango
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why is the probability unchanged, you're not answering that

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induction is unnecessary in any event

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the game starts, 52 cards face down

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the player declares he wants the next card

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it's 50-50 whether the card is black or red

last halo
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@hollow mango Sorry but i'm not getting what you are trying to say.

hollow mango
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you are rephrasing what you have to prove

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in your induction argument

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you are not proving

last halo
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@hollow mango I'm saying that since probability of wining <= 1/2 for n cards by induction hyppthesis. Adding one black and one red card does not change the balance. The ratio of black to red remains same as it was in n cards. Then i claim Since the ratio remains balanced and unaffected by the addition of one black and one red card, no additional information or advantage is introduced. Therefore the probability remains same for n + 2.

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what exactly is the flaw here?

hollow mango
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and if there is no flaw why do you need induction in the first place?

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you have rephrased the thing you Need to prove in equivalent terms

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that does not qualify as proof

last halo
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@hollow mango ?

hollow mango
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but not proving

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regardless: induction is unnecessary

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the rules of the game state that winning/losing is decided with the first card

last halo
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How is it not proving?

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can you be specific ?

last halo
hollow mango
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The cards are face down. Cards are drawn one at a time from the top of the deck and turned face up. At any point before the deck is exhausted, you must declare that you want the next card.

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at the start of the game, the deck is not exhausted

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no cards are turned over

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the player must declare they want a card

hollow mango
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that's essentially what you are doing

hollow mango
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and most importantly

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  1. A card is drawn only when the player declares the draw.
last halo
hollow mango
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im just quoting the 1st post

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if you wanna solve a different problem, then change the problem statement

last halo
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what was the point of qouting ?

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Brother i'm not getting you at all

hollow mango
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that's not what the rules of the game say right now

last halo
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i think you have weak english.

hollow mango
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i think you have weak logic

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why are you arguing with me, if you are so sure your proof is correct why are you asking for validation and get upset when I tell you your proof attempt is circular

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..weak english