Probability of being dealt a blackjack

A two-card blackjack arrives about once every 21 hands: 4.83% of the time from a single deck and 4.75% from a six-deck shoe. The numbers come from counting combinations rather than simulation, and the small gap between them is the first and simplest illustration of why deck count matters in blackjack.

Single deck: 4 aces, 16 tens, 1,326 two-card hands

A blackjack needs one ace and one ten-value card (10, J, Q or K). A standard deck has 4 aces and 16 ten-value cards. The number of distinct two-card hands from 52 cards is:

C(52, 2) = 52 × 51 / 2 = 1,326.

The number of those hands that are an ace plus a ten is 4 × 16 = 64, because any of the 4 aces can pair with any of the 16 tens. So:

P(blackjack, single deck) = 64 / 1,326 = 32/663 ≈ 0.04827, or 4.83%.

The Wizard of Odds works the same figure as a sequence of draws, 2 × (4/52) × (16/51) = 128/2,652 = 0.0482655, where the factor of 2 covers ace-then-ten and ten-then-ace. Both routes give the same answer: about 1 hand in 20.7.

Six decks: the same count with bigger numbers

A six-deck shoe holds 312 cards, including 24 aces and 96 ten-value cards.

C(312, 2) = 312 × 311 / 2 = 48,516.

Blackjack combinations = 24 × 96 = 2,304.

P(blackjack, six decks) = 2,304 / 48,516 = 192/4,043 ≈ 0.04749, or 4.75%.

That is about 1 hand in 21.1. The Wizard of Odds gives the same 192/4,043 and the general formula for n decks, 2 × (4/13) × (4n / (52n − 1)): the first card is an ace or ten with the usual frequency, and the second must be the complementary rank from what remains.

All common deck counts

These are the figures in the site’s data file /data/blackjack-probability.json. The Wizard of Odds probability page lists 4.827%, 4.749% and 4.745% for one, six and eight decks, and Wikipedia’s summary of “approximately 4.8% of hands” is the same figure rounded.

Why more decks mean fewer blackjacks

The difference is entirely a removal effect. Once your first card is an ace, the question is what fraction of the remaining cards are tens:

• Single deck: 16 tens among 51 remaining cards = 31.37%.
• Six decks: 96 tens among 311 remaining cards = 30.87%.
• Infinite deck: 4/13 = 30.77%.

In a single deck, removing one card changes the composition of what is left by about 2% of the pack. In six decks the same card is 0.3% of the pack. The ace you were dealt does not remove any tens, but it does shrink the denominator less in a big shoe, so the ten-rich fraction is slightly lower. The effect is small in absolute terms (0.08 percentage points between one and six decks) but it is the same mechanism that makes single-deck games better for the player in other ways, as the deck count rules article explains.

The dealer also has blackjack: the push probability

When you hold a natural, you have taken one ace and one ten out of play. The dealer’s two cards come from the rest.

Single deck. 50 cards remain with 3 aces and 15 tens. Dealer two-card hands: C(50, 2) = 1,225. Dealer blackjacks: 3 × 15 = 45.

P(dealer blackjack | you have blackjack) = 45 / 1,225 = 9/245 ≈ 3.67%.

Six decks. 310 cards remain with 23 aces and 95 tens. Dealer hands: C(310, 2) = 47,895. Dealer blackjacks: 23 × 95 = 2,185.

P(dealer blackjack | you have blackjack) = 2,185 / 47,895 ≈ 4.56%.

Notice the direction: here the single deck is better for you again, because your natural has removed a larger share of the dealer’s aces and tens. Multiply by your own blackjack probability to get the chance of a double natural on a given hand: about 0.18% in a single deck and 0.22% in six decks, or roughly one round in 460.

Under US peek rules the dealer checks for a natural before anyone plays, so a pushed blackjack is settled immediately. Under European no-hole-card rules the push is discovered at the end of the round; the European rules article covers the difference.

What a blackjack is worth

A natural that is not pushed pays 3:2 under the site’s profile and under any table worth playing: a £10 stake returns £25 (the £10 stake plus £15 profit). Averaged across all hands, the bonus half-unit on winning naturals is worth about:

0.0475 × (1 − 0.0456) × 0.5 ≈ 0.0227, or 2.3% of the stake per round (six decks, illustrative).

That 2.3% is the single largest positive component in the house edge calculation. Cutting the payout to 6:5 removes 0.3 units from each winning natural, which the Wizard of Odds rule-variation table prices at 1.39% of the stake, roughly tripling a typical house edge. Paying even money on blackjacks costs 2.27% in the same source, consistent with the 2.3% estimate above. The 6:5 vs 3:2 article sets out the comparison in full.

Related reading

• Insurance and even money maths
• How the house edge is calculated
• 6:5 blackjack vs 3:2
• Deck count: single vs multi