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

[Quail]

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?

 

[juvenal]

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.

 

[neurocomp2003]

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.

 

[juvenal]

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?

 

[DaveC426913]

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.)