By the bjshoe team · The math behind every blackjack hand · Updated May 2026
Blackjack is one of the most analyzable casino games in existence. Every probability — your bust chance, the dealer's bust chance, the value of each rule variation — can be computed exactly. This page collects the numbers that matter.
This is just the count of remaining cards that would push you over 21, divided by the deck. Approximate values for a 6-deck shoe:
| Your hard total | Bust probability if you hit |
|---|---|
| 11 or lower | 0% |
| 12 | 31% |
| 13 | 39% |
| 14 | 56% |
| 15 | 58% |
| 16 | 62% |
| 17 | 69% |
| 18 | 77% |
| 19 | 85% |
| 20 | 92% |
This is why basic strategy says stand on hard 17+: hitting busts more than two-thirds of the time, and a 17 already loses to most dealer hands anyway.
The dealer's fixed rule (hit until 17+) means we can compute their bust rate from each starting up-card. This is the single most important table for shaping your strategy:
| Dealer up-card | Dealer bust probability |
|---|---|
| 2 | ~35% |
| 3 | ~37% |
| 4 | ~40% |
| 5 | ~42% |
| 6 | ~42% |
| 7 | ~26% |
| 8 | ~24% |
| 9 | ~23% |
| 10 / J / Q / K | ~21% |
| Ace | ~12% |
The pattern: dealer's 4-5-6 up-cards are by far the weakest (~40% bust rate). When the dealer shows one of these, you should stand on stiff totals (12-16) and double down more aggressively. When the dealer shows 7 or higher, they're likely to make a strong hand — you need to hit until you reach 17+.
From a freshly shuffled 6-deck shoe, the probability of being dealt a natural (Ace + 10-value) on your first two cards is approximately 4.75% — or about 1 in 21 hands. Since natural blackjacks pay 3:2, this single statistic accounts for a large share of your overall edge.
Each rule has a known effect on the house's advantage. Positive numbers favor the house; negative numbers favor the player.
| Rule | Effect on house edge |
|---|---|
| Single deck (vs 6 deck) | −0.59% |
| 2 decks (vs 6) | −0.20% |
| 8 decks (vs 6) | +0.02% |
| Dealer hits soft 17 (H17) | +0.22% |
| Blackjack pays 6:5 instead of 3:2 | +1.39% |
| Double after split (DAS) allowed | −0.14% |
| Resplit Aces (RSA) allowed | −0.08% |
| Late surrender allowed | −0.07% |
| Early surrender allowed | −0.62% |
| Double restricted to 9–11 only | +0.09% |
| No double after split | +0.14% |
| European No-Hole-Card rule | +0.11% |
EV is what you'd average if you made the same decision over an infinite number of identical scenarios. A few classic examples:
Most casino games have a fixed house edge baked into the bet (roulette ~2.7%, single-zero; baccarat ~1.06% banker; craps pass line ~1.41%). Blackjack is different: the dealer also has to play out their hand against the same shoe you do, and crucially, you go first. That means if you bust before the dealer plays, you lose immediately even if the dealer would have busted too. That asymmetry is most of the house's edge. Player advantages — 3:2 blackjack payouts, doubling, splitting, surrender — claw most of it back.
A 0.55% house edge means that over the long run, for every $100 you wager (cumulatively), you expect to lose about 55 cents. That's the long-run average. Short-term variance is huge: you can win or lose dozens of bets in a row by pure luck. Bankroll management, not edge alone, determines whether you walk away ahead in any given session.
▶ Play bjshoe — practice with the right rules, free