# Why a Royal Flush Beats Quad Aces

> Why a royal flush beats quad aces, with the Mabuchi–Phillips hand, scoped two-player odds and an exact turn showdown equity calculation.

- Source: https://toolsgambling.com/blog/quad-aces-vs-royal-flush
- Author: Evgeniy Volkov (ToolsGambling)
- Published: 2026-02-25
- Updated: 2026-10-08
- Cite as: "Quad Aces vs Royal Flush: Which Hand Wins?", ToolsGambling, https://toolsgambling.com/blog/quad-aces-vs-royal-flush

**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](https://www.pokerstars.com/poker/games/rules/hand-rankings/).

## 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](https://www.espn.com/espn/poker/columns/story?columnist=lee_bernard&id=4825413) and [the official WSOP recap](https://www.wsop.com/news/wsop-highlights-powered-by-pokergo-2008-wsop-main-event-top-5-hands/).

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

| Street | Board | What changed |
|---|---|---|
| Flop | A♥ Q♦ 9♠ | Mabuchi had three aces |
| Turn | A♥ Q♦ 9♠ T♦ | Phillips already had a king-high straight: 9♠ T♦ J♦ Q♦ K♦ |
| River | A♥ Q♦ 9♠ T♦ A♦ | Mabuchi had four aces; Phillips had a royal flush |

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](https://wizardofodds.com/ask-the-wizard/215/) 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.

| River cards | Count | Result |
|---|---:|---|
| Q♠ Q♥ Q♣ | 3 | Mabuchi makes a full house and beats Phillips’s straight |
| T♠ T♥ T♣ | 3 | Mabuchi makes a full house and beats Phillips’s straight |
| 9♥ 9♣ | 2 | Mabuchi makes a full house and beats Phillips’s straight |
| All other cards | 36 | Phillips wins |

**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](https://toolsgambling.com/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♦](https://toolsgambling.com/media/article/2026/10/68d456fa3376d35807d08bf6ac652c43.webp)

*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.

| Hand | Probability |
|---|---:|
| Royal flush | 0.000154% |
| Straight flush excluding royal flushes | 0.00139% |
| Four of a kind | 0.024% |
| Full house | 0.144% |
| Flush | 0.197% |

**Hand probabilities in a random five-card hand**

|  | Probability, % (logarithmic scale) |
|---|---:|
| Royal Flush | 0.000154 |
| Straight Flush excluding Royal Flushes | 0.00139 |
| Four of a Kind | 0.024 |
| Full House | 0.144 |
| Flush | 0.197 |

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.

## FAQ

### Which is stronger: four aces or a royal flush?

A royal flush beats four aces. In standard poker, every straight flush outranks every four of a kind. A royal flush is the ace-high straight flush.

### How often do quad aces face 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. This is not an estimate for a nine-player table.

### What was Mabuchi’s equity on the turn?

With A♠ A♣ against K♦ J♦ on A♥ Q♦ 9♠ T♦, Mabuchi’s showdown equity was 18.2% and Phillips’s was 81.8%. Mabuchi won on eight of the 44 possible rivers, with no ties.

### Why didn’t A♦ give Mabuchi the winning hand?

A♦ gave Mabuchi four aces, but it also completed Phillips’s diamond royal flush: T♦ J♦ Q♦ K♦ A♦. The royal flush outranks four of a kind.

### Does the equity calculation prove that an all-in was correct?

No. Showdown equity measures a player’s share of the pot under specified cards and runouts. The calculator does not model fold equity, pot odds or the expected value of a decision.