The second Ace

Probability Level 4

You have seven cards in your hand in a random order, and two are Aces.

You flip them over one at a time until you have flipped over both Aces.

The expected number of flips is a b \frac{a}{b} where a a and b b are coprime positive integers.

What is a + b a+b ?

Image credit: s.hswstatic.com


The answer is 19.

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Nov 1, 2018

There are ( 7 2 ) = 21 \binom{7}{2} = 21 possibilities for the positions of the Aces.

  • 6 of them result in the second Ace being in the 7th position.
  • 5 of them result in the second Ace being in the 6th position.
  • 4 of them result in the second Ace being in the 5th position.
  • 3 of them result in the second Ace being in the 4th position.
  • 2 of them result in the second Ace being in the 3th position.
  • 1 of them results in the second Ace being in the 2nd position.

So,

E = 1 ⋅ 2 + 2 ⋅ 3 + 3 ⋅ 4 + 4 ⋅ 5 + 5 ⋅ 6 + 6 ⋅ 7 21 = 16 3 E = \dfrac{1\cdot2 +2\cdot3 +3\cdot4 +4\cdot5 +5\cdot6 +6\cdot7}{21} = \dfrac{16}{3}

16 + 3 = 19 16 + 3 = \boxed{19}

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