Deriving Basic Strategy

Basic strategy looks like a magic table, but every cell comes from one simple principle: take the action with the highest expected value. For each hand you compare the EV of standing, hitting, doubling, and splitting, and pick the best. The table is just the answer key to millions of such comparisons.

A taste of the computation

With an infinite-deck approximation, every card value is equally likely (aces counted as $1$ when they would bust you), and we can already compute useful quantities — for example, the probability of busting when hitting a hard total.

The code below is written as it would appear in a Jupyter notebook cell:

cards = [2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 10, 10] + [1]  # ace = 1

def p_bust(hard_total):
    return sum(c > 21 - hard_total for c in cards) / len(cards)

for s in range(12, 21):
    print(f"hard {s}: p(bust) = {p_bust(s):.3f}")

Running it shows why hitting $16$ is so painful:

hard 12: p(bust) = 0.308
...
hard 16: p(bust) = 0.615

From bust probabilities to strategy

Bust probabilities alone are not enough — standing wins whenever the dealer busts. The full derivation compares, for a fixed player total and dealer upcard,

$\mathrm{EV}(\text{hit}) \quad \text{vs} \quad \mathrm{EV}(\text{stand}),$

each computed by summing over all dealer outcomes. Do this for all $550$ or so combinations and the classic strategy table pops out. In future posts we will build this computation up properly.