You hold a hard 9 (no ace counted as 11). The dealer shows a 3. Here is the mathematically correct basic-strategy play — and why.
| Double ← | +0.1208 |
| Hit | +0.1013 |
| Stand | -0.2522 |
| You bust on the next card | 0.0% |
| Dealer 3 busts | 37.4% |
Double beats Hit by 0.0196 per unit staked. Small, but it is the same small edge in every shoe you ever play — and it compounds over a lifetime of hands. Across the 340 situations in this chart it ranks 127th by expected value — better than 213 of them, worse than 126. Expected value is per unit staked, computed for multi-deck, dealer stands on soft 17, double after split allowed. The model and the code are published as data.
Hard 9 is a premium doubling hand. One card is very likely to give you a strong total, and against a 3 (which busts 37% of the time) you want maximum money on the table while you have the edge. Doubling wins you more in exactly the spots where you're a favourite.
If doubling isn't allowed (e.g. after hitting), simply hit instead — the extra card logic stays the same, you just can't add the extra bet.