The Ace of Spades

Probability Level 4

I have a standard deck of 51 cards, with one missing Ace. I shuffled the cards and the cards then turned up one by one from the top until the second ace appears. What is the expected number of cards to be turned up?

This problem is adapted from a past USAMO.


The answer is 26.

Christopher Boo by Christopher Boo , 24, Singapore

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1 solution

Jack D'Aurizio
Apr 15, 2014

Symmetry is the key. If the second ace appears in position a a , by reversing the deck the second ace appears in position 52 − a 52-a , hence the expected value is just 26 26 .

5 Helpful 0 Interesting 0 Brilliant 0 Confused

E please explain.

Adarsh Kumar - 7 years, 1 month ago

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I will try to be more clear. A deck and its reversed deck have the same probability, and for such a couple the sum of the positions of the second ace is always 52. By averaging over all the deck/reversed deck couples, i.e. by averaging over the whole set of decks, we get that the expected position of the middle ace is just 26, i.e. the middle position.

Jack D'Aurizio - 7 years, 1 month ago

Yeah could explain further please?

Maham Zaidi - 7 years, 1 month ago

its practically wrong sci means experimantly truth

Rohit Singh - 7 years, 1 month ago

but aren't there four ace's initially in a normal deck of card? I think you are assuming that there are only 2 aces in a deck of card.

Park Sejin - 5 years, 7 months ago

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