Probability Your Card Beats the Dealer’s

You draw one card from a standard 52-card deck. Then the dealer draws one card without replacement. You win only if your card’s rank is strictly higher.

Final Answer:
8/17 ≈ 47.06%

Win

8/17
≈ 47.06%

Tie

1/17
≈ 5.88%

Lose

8/17
≈ 47.06%

Visual Breakdown

By symmetry, excluding ties, you and the dealer are equally likely to have the higher rank. The only catch is that ties count as losses for you.

Win 47.06%
Tie 5.88%
Lose 47.06%

Method 1: Symmetry + Ties

  1. After you draw a card, there are 51 cards left.
  2. Exactly 3 of them have the same rank as yours, so
    P(tie) = 3 / 51 = 1 / 17
  3. Therefore,
    P(non-tie) = 1 - 1/17 = 16/17
  4. On all non-ties, symmetry gives:
    P(you higher | non-tie) = 1/2
  5. So,
    P(win) = (1/2) × (16/17) = 8/17

Method 2: Count Ordered Pairs

  1. Total ordered pairs of draws:
    52 × 51 = 2652
  2. Tie pairs: choose the rank in 13 ways, your suit in 4 ways, dealer suit in 3 ways:
    13 × 4 × 3 = 156
  3. Non-tie pairs:
    2652 - 156 = 2496
  4. Half of the non-ties are wins for you:
    2496 / 2 = 1248
  5. Hence,
    P(win) = 1248 / 2652 = 8 / 17

Key Insight

If ties did not exist, the game would be exactly 50-50 by symmetry. But ties do exist, and they happen with probability 1/17. Since ties count against you, your winning probability becomes:

1/2 × (1 - 1/17) = 1/2 × 16/17 = 8/17

So the probability that your card beats the dealer’s is 8/17, approximately 47.06%.