One card each in casino war

You and the dealer are each dealt one card from a fair, shuffled 5252-card deck. You win only if your card's rank is strictly higher than the dealer's. What is the probability you win?

Show hints (2)+
  1. Win and lose are symmetric: 2P(win)+P(tie)=12\,\mathbb{P}(\text{win})+\mathbb{P}(\text{tie})=1.
  2. A tie is matching ranks: given your card, 3/513/51 of the rest match.

Answer

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0.4706 (± 0.001)

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Asked at: Jane Street, Optiver

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