Why a Royal Flush Beats Quad Aces

A royal flush beats four aces in standard poker. Every straight flush—five consecutive cards of the same suit—outranks four of a kind.

Four of a kind consists of four cards of the same rank. A royal flush is the highest straight flush: ten, jack, queen, king and ace of one suit. At showdown, the hand category takes priority over the ranks within it. Having four aces does not make four of a kind stronger than a straight flush; see PokerStars’ official hand-ranking rules.

How Mabuchi Lost with Four Aces

In the verified 2008 hand, the river gave Mabuchi four aces while completing Phillips’s diamond royal flush; see ESPN’s report on the hand and the official WSOP recap.

Mabuchi held A♠ A♣, and Phillips held K♦ J♦. The community cards arrived in this order:

At showdown, Mabuchi could use A♠ A♣ A♥ A♦ for four of a kind. Phillips could use T♦ J♦ Q♦ K♦ A♦ for a royal flush, which won the hand.

Phillips did not need the river to take the lead. His straight on the turn already beat Mabuchi’s three aces.

What Is the Probability of Quad Aces Against a Royal Flush?

In the specified random two-player Texas Hold’em deal model, the probability is 0.000000607468%, or about 1 in 165 million.

The model assigns two hole cards to the first player, two to the second and five community cards. Card order within each group is ignored:

• Total deals: C(52,2) × C(50,2) × C(48,5) = 2,781,381,002,400.
• Favorable deals: 16,896 = 2 × (3,168 + 3,168 + 2,112).
• Probability: 6.0746801626e-9 = 0.000000607468%.
• Frequency: 1 in 164,617,720.3, rounded to about 1 in 165 million.

Here, C(n,k) means the number of ways to choose k cards from n without regard to order. The factor of two accounts for reversed player roles: either player can hold the four aces.

Wizard of Odds’ “Ask the Wizard” #215 analysis supplies the case counts and explicitly says to reverse the player roles. Its displayed equation and result are inconsistent, so the values above use the independently recomputed total of 16,896 favorable deals.

This probability applies to that specific deal model. It is neither a nine-player-table estimate nor the probability of reproducing Mabuchi and Phillips’s exact cards. ESPN’s quoted figure of 2.7 billion is not supported by this calculation; its source or method should not be inferred from it.

Which River Cards Could Have Saved Mabuchi?

Mabuchi won on 8 of the 44 possible river cards: three queens, three tens and two nines. His showdown equity was 18.2%, compared with 81.8% for Phillips.

This calculation fixes both players’ hole cards and the turn board, A♥ Q♦ 9♠ T♦. Eight cards are known, leaving 44 possible one-card runouts, each treated as equally likely.

A♦ was not a winning out for Mabuchi. It improved his hand to four of a kind but gave Phillips a royal flush. 9♦ also failed to save Mabuchi: Phillips would make the straight flush 9♦ T♦ J♦ Q♦ K♦, beating Mabuchi’s full house.

There are no ties in this calculation, so each player’s showdown equity numerically equals their win percentage. That is specific to this matchup, not a general equivalence: equity includes a player’s share of tied pots, whereas win percentage counts outright wins. The displayed values, 18.2% and 81.8%, are rounded to one decimal place.

How to Check the Hand in the Calculator

You can reproduce the result by entering the two fixed hands and leaving the river empty.

1. Open the poker equity calculator.
2. Set the first player’s hole cards to A♠ A♣ and the second player’s to K♦ J♦.
3. Set the board to A♥ Q♦ 9♠ T♦, leaving the river empty.
4. Run the calculation. If you change any cards, run it again.
5. Check the result: exact enumeration covers 44 runouts, with 8 wins for Mabuchi and 36 for Phillips.

Poker equity calculator showing Mabuchi’s A♠ A♣ against Phillips’s K♦ J♦ on the turn board A♥ Q♦ 9♠ T♦

Capture of the real calculator running on localhost, not production. “Hero” represents Mabuchi and “Villain” represents Phillips. The river is empty so the calculation includes all 44 possible runouts.

The calculator supports fixed cards and hand ranges. Use fixed cards here to reproduce this exact matchup. The tool was verified locally; verification of the production page was blocked by an HTTP 403 response.

The result measures showdown equity, not the expected value of a decision. The calculator does not model fold equity, pot odds, decision EV or jackpot qualifications. These percentages alone therefore do not establish whether an all-in—a bet of all remaining chips—was correct.

How Rare Are These Hands in a Random Five-Card Hand?

The probabilities below apply to one random five-card hand from a standard deck, not to the seven cards available to a Texas Hold’em player.

The chart compares five categories. It lists a royal flush separately, so the straight flush entry excludes royal flushes.

For comparison, a straight has a probability of 0.392% in the same random five-card-hand model. These rounded figures describe individual hand frequencies, not seven-card Hold’em probabilities or player matchup odds. Quad aces facing a royal flush requires the separate two-player deal calculation given above.