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# Dueling cards

## Question

Moe and Joe had some time on their hands, so they played a simple game called "Dueling Cards."

In this game, each player has a deck of cards, shuffled and without jokers.

A hand consists of a single card taken from the top of each player's deck.

Suits and face values are both ranked. No two cards in a deck have the same rank.

A player wins the hand by holding a card that outranks the card of his opponent.

The bank pays 1 chip to the winner of a hand.

It's a simple game. No poker hands. No betting. Just high card wins.

After 52 hands the decks are re-shuffled, and play continues as long as desired.

Moe and Joe played through their decks 10 times, then stopped to see who won.

As they counted their chips Moe (a statistician) said, I wonder what the most likely winning score is.

Joe (a simpler person) replied, I wonder what the most likely combined score is.

I'm sure Bushindo can answer Moe's question.

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## Recommended Posts

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510

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Does "combined score" mean the total number of chips Moe and Joe have together after 10 rounds?

If so, then the answer is 520, but where is the puzzle?

I'm sure that's not the right answer, so I'm not hiding it in the spoiler.

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Does "combined score" mean the total number of chips Moe and Joe have together after 10 rounds?

If so, then the answer is 520, but where is the puzzle?

I'm sure that's not the right answer, so I'm not hiding it in the spoiler.

520 is not the answer because neither Joe nor Moe gets a chip if they turn over cards of the same rank (which, you'll have to admit, will sometimes happen).

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Does "combined score" mean the total number of chips Moe and Joe have together after 10 rounds?

If so, then the answer is 520, but where is the puzzle?

I'm sure that's not the right answer, so I'm not hiding it in the spoiler.

520 is not the answer because neither Joe nor Moe gets a chip if they turn over cards of the same rank (which, you'll have to admit, will sometimes happen).

I knew I was missing something obvious.

"No two cards in the deck have the same rank" is what tripped me into thinking that ties are not possible. I agree with your answer of 510 as the expected number of ties in each game is 1.

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• 0

Does "combined score" mean the total number of chips Moe and Joe have together after 10 rounds?

If so, then the answer is 520, but where is the puzzle?

I'm sure that's not the right answer, so I'm not hiding it in the spoiler.

520 is not the answer because neither Joe nor Moe gets a chip if they turn over cards of the same rank (which, you'll have to admit, will sometimes happen).

I knew I was missing something obvious.

"No two cards in the deck have the same rank" is what tripped me into thinking that ties are not possible. I agree with your answer of 510 as the expected number of ties in each game is 1.

I'll take slight credit for purposefully obfuscating that precise point (no two cards in a deck have ...) Heh.

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