Who's Right in this Bet: 1 in 5 or 1 in 50?

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Discussion Overview

The discussion revolves around a probability bet involving a deck of 50 cards, where 1 card is the ace of spades. Participants debate whether the chance of turning over the ace of spades is 1 in 5 or 1 in 50 when the cards are dealt into 5 stacks of 10 cards each.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant argues that the probability remains 1 in 50 regardless of how the cards are stacked, stating that subdividing the stacks does not change the overall probability.
  • Another participant suggests that if the ace of spades is in a stack, the probability could be viewed as 1 in 5, depending on how one interprets the situation.
  • A different viewpoint emphasizes that redefining the game or the conditions alters the probability, questioning the validity of changing the stack sizes.
  • One participant raises a question about whether the friend can see any cards before attempting to turn over the ace of spades, suggesting that this could affect the odds.
  • Another participant challenges the friend's reasoning by proposing a hypothetical scenario where the probabilities would exceed 100%, indicating a flaw in the friend's logic.

Areas of Agreement / Disagreement

Participants express differing views on the probability of turning over the ace of spades, with no consensus reached. Some maintain that it is 1 in 50, while others argue for a 1 in 5 probability under certain interpretations.

Contextual Notes

Participants highlight the importance of definitions and interpretations in probability, indicating that assumptions about visibility of cards and the structure of the stacks may influence the discussion.

Quail
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I have a bet with a friend who says that if you have 50 cards with 1 card being the ace of spades which you deal in 5 10 card stacks you then get a 1 in 5 chance of turning the ace of spades instead of a 1 in 50. He gets there by making 5 one card stacks with 1 stack being the ace of spades which would be 1 in 5. He then puts the remainder of the 50 cards in each stack creating 10 cards per stack and argues the chances of turning the ace of spades remains a 1 in 5 chance. I insist that it goes to a 1 in 50 chance. Who is correct and why?
 
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Your probability doesn't change no matter how you might subdivide the stacks. That is irrelevant. It remains 1/50.

A simple argument to see this is proof by contradiction: The probabilities must all add to one. Let's assume that every card is different and that the deck is only missing the two of clubs and the two of hearts.

If your friend were correct, then the probability of getting an ace, king, queen, jack, 10, or 9 of spades is equal to 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5 = 6/5 which is already greater than 100%.

In fact the total probability (getting any card) would be 50/5 = 10. That's impossible.
 
well it depends...on how your viewing it...if you can turn over all cards in a stack and say
the AoS is there...then yes it is 1/5 because your space has changed into the stacks and no the cards...so it depends on how you view it.
 
That's redefining the game then.

Look, I could create a game where the stack is size one. Turn over the "stack" and if I have an ace of spades, then my probability of getting an ace of spades is 100% for that "stack". What's the point?
 
Quail said:
I have a bet with a friend who says that if you have 50 cards with 1 card being the ace of spades which you deal in 5 10 card stacks you then get a 1 in 5 chance of turning the ace of spades instead of a 1 in 50. He gets there by making 5 one card stacks with 1 stack being the ace of spades which would be 1 in 5. He then puts the remainder of the 50 cards in each stack creating 10 cards per stack and argues the chances of turning the ace of spades remains a 1 in 5 chance. I insist that it goes to a 1 in 50 chance. Who is correct and why?

One question: does his solution allow him to see any cards before he makes his 1-in-5 try at turning up the AoS? If so, he changes the odds. If not, it's 1-in-50.

Forget logic - have him prove it. Have him subdivide the deck any way he wants. Get him to turn up the Ace of Spades one out of five times. (If he does, I'd take out a second mortgage on his career as a magician.)
 

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