A deck holds 52 cards in 13 ranks, four of each. Blackjack scores ten, jack, queen and king the same, so what looks like a thirteen-way game is really a ten-way one with an uneven split. Nine values (ace to nine) have four cards each. One value has sixteen. The shoe in Blackjack 21 Big Cards Offline holds six decks, 312 cards, and the proportions are identical: 96 tens and 24 of everything else.
The share of each value
Sixteen cards in 52 is 16/52 = 30.77%. Each of the other nine values is 4/52 = 7.69%. A ten-value card is exactly four times as likely as any other particular value, and slightly under one card in three overall.
Once cards leave the pack the shares drift a little, but only a little. With one ten gone, a six-deck shoe has 95 tens in 311 cards, 30.55%. The six-deck versus single-deck post covers how fast that drift grows in a smaller pack.
The hole card is probably a ten, and probably not
When the dealer shows a six, a good player thinks of the dealer as holding sixteen: a ten is the most likely hole card. That is a fair rule of thumb, because no other value comes close. But it is the likeliest single value, not a likely one. The hole card is a ten 30.77% of the time and something else 69.23% of the time.
The same count decides what the dealer's peek finds. With an ace showing, 96 of the 311 unseen cards are tens, so the dealer has a natural 30.87% of the time (counting only the ace you can see). With a ten showing, 24 of 311 are aces, a natural 7.72% of the time. The two chances differ by a factor of four, the same factor as the chart above.
Why a two-card 20 is common
A hand of two ten-value cards needs the first card to be a ten, 96/312, and the second to be one of the 95 left among 311 cards:
96/312 × 95/311 = 9.40%
Add the other way to make 20 in two cards, a soft 20 of ace and nine, at 2 × 24/312 × 24/311 = 1.19%, and about one hand in 9.4 starts on 20 (10.59%). Just over half of all two-card hands, 52.14%, contain at least one ten-value card.
The hard 20 of two tens alone is almost exactly twice as likely as a natural. A natural, an ace and a ten, comes to 2 × 24/312 × 96/311 = 4.75%, one hand in 21.06. A natural needs a ten and an ace, and the ace is only 7.69% of the deck. If tens were a single rank like the rest, with 24 cards in the shoe, a natural would arrive 2 × 24/312 × 24/311 = 1.19% of the time. The extra tens make naturals four times as frequent. The 3:2 payout post shows what that frequency is worth and why the rate of naturals matters more than any other rule.
Why 12 to 16 are stiff hands
A stiff hand is a hard total from 12 to 16, where one more card can bust you but cannot yet make a strong hand. The reason these hands are bad follows from the tens. In 13 ranks, a hit busts a hard total of t when the card is at least 22 minus t, and the ten-value cards are always among them, worth four ranks on their own. Counting by rank, with no cards removed:
- Hard 12: only a ten-value card busts, 4 ranks of 13, 30.77%.
- Hard 13: a nine or any ten, 5 of 13, 38.46%.
- Hard 14: an eight, a nine or any ten, 6 of 13, 46.15%.
- Hard 15: a seven, an eight, a nine or any ten, 7 of 13, 53.85%.
- Hard 16: a six, seven, eight, nine or any ten, 8 of 13, 61.54%.
Hard 17 busts 9 of 13, 69.23%, which is why nobody hits it. Tens are the largest piece of every bust figure: all of the risk at 12, and half of it at 16. Meanwhile the cards that actually help a stiff hand are the small ones, and they are each worth only 7.69%.
The reverse happens to the dealer, who must draw to 16. A dealer six with a ten underneath, 30.77% of the time, is on 16 and has to hit it. This is the mechanism behind the bust rates in the post on how the dealer plays: a 6 busts 42.3% of the time and a 10 only 21.2%.
You do not need to memorise a bust table. Take a hard 16: 8 ranks of 13 bust, and four of those eight are tens. Count the ten-value cards and the rest follows.
What to take to the table
Three habits come out of the numbers. Treat the hole card as a ten when planning a hand, while remembering that it is wrong more than two times in three. Expect to start on 20 about once in every nine or ten hands, and stand without a second thought. And when you hold a hard 12 to 16 you are not facing a one-in-three danger, you are facing 31% to 62% depending on the total, with tens doing most of the damage.
The practice table in Blackjack 21 Big Cards Offline is a six-deck shoe with blackjack paying 3:2 and the dealer standing on 17, so these are the proportions you are dealt from.
Blackjack 21 Big Cards Offline deals from a six-deck shoe with blackjack paying 3:2 and the dealer standing on 17, and its How to play guide covers card values, every move, dealer rules and payouts. It plays offline, the chips are free every day, and they have no cash value.