The chance on a single hand
A natural is an ace and a ten-value card (a ten, jack, queen or king) as the first two cards. With a single 52-card deck the chance can be worked out directly. There are 4 aces and 16 ten-value cards, and either one can come first:
- Ace first, ten-value second: 4/52 x 16/51
- Ten-value first, ace second: 16/52 x 4/51
Adding the two gives 128/2652, which is about 0.0483, or 4.8%. That is roughly 1 hand in 21.
With many decks the figure drops slightly, because removing one card from a big shoe changes the remaining probabilities less. With very large numbers of decks the chance approaches 2 x (1/13) x (4/13) = 8/169, about 4.7%. So the natural frequency at a real table is typically a little under 4.8%, depending on the number of decks. This article uses 4.8% as a round, illustrative figure.
Expected naturals in a session
The expected count is simply the number of hands multiplied by the chance per hand:
| Hands | Expected naturals (at 4.8%) |
|---|---|
| 20 | about 1 |
| 100 | about 4.8 |
| 500 | about 24 |
| 1,000 | about 48 |
Expected means average over many sessions, not what you will see in any single one. Over 100 hands you would not be surprised by 2 naturals or by 8.
The chance of seeing at least one
It is often more useful to ask how likely it is to see at least one natural in a given number of hands. The easiest way is to find the chance of seeing none, which is 0.952 (1 minus 0.048) multiplied by itself once per hand, and subtract from 1.
| Hands | Chance of no natural | Chance of at least one |
|---|---|---|
| 10 | about 61% | about 39% |
| 20 | about 37% | about 63% |
| 50 | about 9% | about 91% |
| 100 | under 1% | over 99% |
In other words, a 20-hand stretch without a natural is a fairly ordinary event, and happens a little over a third of the time. Even 40 hands without one happens roughly one session in seven (0.952 to the 40th power is about 14%). Waiting on a natural is not a sign that the shoe is unfavourable, and a natural just seen does not make the next one less likely. Believing otherwise is the gamblers fallacy.
How much the count varies
The number of naturals in a set of hands follows an approximately binomial pattern. The spread is described by the standard deviation, which here is the square root of hands x 0.048 x 0.952:
- Over 100 hands: expected 4.8, standard deviation about 2.1. A count between roughly 3 and 7 is typical, and counts outside that range happen from time to time.
- Over 1,000 hands: expected about 48, standard deviation about 6.8. A count between roughly 41 and 55 is typical.
As the number of hands rises, the count as a share of hands gets closer to 4.8%, even though the raw count wanders further from its expected value in absolute terms. This is the law of large numbers in a small setting.
What a natural is worth, and what it is not
A natural usually pays 3 to 2, so a 10-unit bet wins 15 units, though some tables pay 6 to 5, which would win 12 units on the same bet. The payout is a large part of the player’s side of the equation, which is why natural payouts differ so much in value from table to table.
A natural is also not certain to win. If the dealer shows an ace or a ten-value card, the dealer can also have a natural, and then the hand is a push. The dealer natural is a separate calculation, because the upcard changes the odds. Offers of even money when you hold a natural and the dealer shows an ace are a related decision.
It is also important not to read the natural rate as a route to profit. The frequency is fixed by the cards, so there is nothing to be gained by waiting for them or by changing play to chase them, and the house edge applies to the whole game, naturals included. The 4.8% figure also applies to the player’s hands in an ordinary deal: it says nothing about when the next one will arrive.
Session pace
The number of hands in a session depends on the speed of the game. Online and automated games can deal many more hands per hour than a busy live table, and so produce more naturals per hour of play, along with more of everything else, including losses. Hands per hour matters as much as the per-hand figures when estimating what a session will look like.
Checking the arithmetic
The one-deck calculation (128/2652 is about 0.0483) and the repeated-multiplication method for “none in n hands” are the same tools you could use for any rare event in cards. They assume independent hands, which is a small simplification, since hands dealt from the same shoe are linked slightly by the cards already seen.
As always, gambling involves risk. These figures help explain what to expect, not how to win. Keep to a budget you can afford and seek help from a local support service if gambling becomes a problem.