Expected Value in Blackjack: The Math Behind Every Decision

Expected value (EV) is the single most important concept in advantage gambling. If you understand EV, you understand why card counting works, why basic strategy is what it is, and why some bets are profitable while others are certain losers.

This post builds EV from first principles, then applies it to blackjack decisions.

What Expected Value Actually Means

Expected value is the average outcome of a bet if you made it infinite times.

For a coin flip betting €1: - 50% chance: win €1 - 50% chance: lose €1 - EV = (0.5 × €1) + (0.5 × -€1) = €0

The EV is zero. This is a "fair" bet—neither side has an advantage.

For roulette betting €1 on red (European single-zero): - 18/37 chance: win €1 - 19/37 chance: lose €1 - EV = (18/37 × €1) + (19/37 × -€1) = -€0.027

The EV is -€0.027, or -2.7%. For every €1 bet, you expect to lose 2.7 cents on average. The house has an edge.

EV in Blackjack: More Complex, Same Principle

Blackjack EV calculation is harder because:

1. Multiple possible outcomes (win, lose, push, blackjack)
2. Multiple possible actions (hit, stand, double, split)
3. Probabilities that depend on visible cards and deck composition

Let's calculate the EV of standing on hard 16 vs. dealer showing 10.

Assumptions: 6-deck shoe, neutral count, basic strategy for all subsequent decisions.

The dealer's hole card probabilities (for a 10 showing): - Dealer has 20: ~31% (any 10-value) - Dealer has 21 (blackjack): ~8% (Ace underneath) - Dealer has 17-19: ~25% (7, 8, 9 underneath) - Dealer must draw: ~36% (2-6 underneath)

If dealer must draw from a stiff hand (12-16), dealer bust probability is ~40%.

Working through all scenarios:

Total EV (stand on 16 vs 10) ≈ -0.54

You expect to lose 54% of your bet on average. That sounds terrible—but it might still be better than hitting.

Comparing Actions: Stand vs Hit

Now let's calculate hitting on 16 vs 10.

If you hit: - Draw 2-5: Improve to 18-21 (keep playing optimally from there) - Draw 6: Get 22, bust immediately - Draw 7: Get 23, bust immediately - Draw 8: Get 24, bust immediately - Draw 9: Get 25, bust immediately - Draw 10-K: Get 26, bust immediately - Draw A: Get 17, then play optimally

Bust probability on 16: hitting draws a 6-K about 62% of the time → immediate -1 EV contribution = -0.62

The remaining 38% continues playing, with further EV calculations needed for each resulting hand.

Full calculation (which requires recursion for multi-card draws):

Total EV (hit on 16 vs 10) ≈ -0.54

Nearly identical! This is one of the closest decisions in blackjack. Basic strategy says hit because the EV is marginally better, but we're talking about fractions of a percent.

Why Count Changes These Decisions

Here's where it gets interesting.

At true count 0, the deck is neutral. Hit and stand are nearly equal on 16 vs 10.

At true count +2: - More 10s remain in the deck - Hitting becomes worse (higher bust probability) - But dealer is also more likely to bust - Net effect: Standing becomes slightly better

At true count -2: - Fewer 10s remain - Hitting becomes safer - Dealer is less likely to bust on stiff hands - Net effect: Hitting becomes clearly better

The "index" for 16 vs 10 is 0. At TC ≥ 0, stand. At TC < 0, hit.

This is how counting systems generate strategy deviations. The EV calculation shifts with deck composition.

Calculating EV: The Recursive Challenge

True EV calculation requires considering all possible future cards and decisions. For "should I hit this 14 vs 10?":

EV(hit 14 vs 10) = 
    P(draw 2) × EV(16 vs 10, best play) +
    P(draw 3) × EV(17 vs 10, best play) +
    P(draw 4) × EV(18 vs 10, best play) +
    P(draw 5) × EV(19 vs 10, best play) +
    P(draw 6) × EV(20 vs 10, best play) +
    P(draw 7) × EV(21 vs 10, best play) +
    P(draw 8+) × (-1)  // bust

Each of those EV(X vs 10) terms requires its own calculation. The recursion continues until all branches reach terminal states (bust, stand, or 21).

For splits, it's worse. Splitting creates two hands, each with their own recursive tree, and you might split again.

This is why basic strategy was computed by mathematicians with mainframes in the 1960s. Today, we can do it on a laptop in milliseconds, but the principle is the same: exhaustive recursive EV calculation.

Doubling Down: When EV Justifies Risk

Doubling means betting twice as much for exactly one card. This is only worthwhile when:

EV(double) > 2 × EV(best alternative)

Example: 11 vs 6 (dealer showing)

If you hit and play optimally, your EV might be +0.30 (you're favored).

If you double, you get one card: - Draw 10: 21 vs dealer's 6 → very strong position - Draw 7-9: 18-20 → strong position - Draw A-6: 12-17 → weaker, but dealer still has stiff hand

Calculated double EV: ~+0.65

Since 0.65 > 2 × 0.30, doubling is correct. You're giving up flexibility (only one card) for the ability to bet more when you're favored.

At negative counts (fewer 10s), the double becomes less attractive because you're less likely to draw 10-A.

Insurance: The Classic EV Trap

Insurance is offered when dealer shows Ace. You can bet half your original bet that dealer has blackjack (10 underneath). Pays 2:1.

EV calculation: - Probability dealer has 10: 4/13 = 30.77% (neutral deck) - Win 2:1 on half-bet: +1 unit when dealer has blackjack - Lose half-bet: -0.5 units when dealer doesn't

EV = (0.3077 × 1) + (0.6923 × -0.5) = 0.3077 - 0.3462 = -0.0385

Insurance has -3.85% EV with a neutral deck. That's worse than most casino games.

But at high counts: - True count +3: roughly 33.8% of remaining cards are 10s - EV = (0.338 × 1) + (0.662 × -0.5) = 0.338 - 0.331 = +0.007

At TC +3, insurance becomes slightly profitable. This is one of the most valuable index plays because you're getting 2:1 on a near coin-flip.

EV and Bet Sizing: The Kelly Connection

Knowing you have positive EV isn't enough. You need to size bets appropriately.

The Kelly Criterion provides the optimal bet sizing:

Optimal bet = (edge / variance) × bankroll

For blackjack: - Edge at TC +4: roughly 1.5% - Variance per hand: roughly 1.3 (higher than simple win/lose due to doubles, splits, blackjacks) - Kelly bet: (0.015 / 1.3) × bankroll ≈ 1.15% of bankroll

With a €10,000 bankroll at TC +4, Kelly suggests betting €115.

Most advantage players use "half Kelly" or "quarter Kelly" to reduce variance at the cost of slower growth. The math is unforgiving—full Kelly has dramatic swings.

Why Negative EV Bets Can Be Correct

Counterintuitive fact: sometimes you should make negative EV bets.

Hitting 16 vs 10 has negative EV. You still do it because standing is more negative.

Similarly, playing at TC 0 has negative EV (house edge ~0.5%). Professional counters still play these hands because:

1. Leaving at TC 0 looks suspicious
2. The time at TC 0 is the "cost" of being present when TC goes high
3. Cover plays maintain access to profitable situations

The goal isn't maximizing EV on every individual bet. It's maximizing total EV across all play, including camouflage.

The Long-Run Reality

EV is an average. Individual sessions will vary wildly.

Even with a 1% edge, you'll have losing sessions. Lots of them. The math says you're favored, but variance doesn't care about the math on any single night.

Standard deviation in blackjack is roughly 1.1 units per hand. Over 100 hands with 1% edge:

• Expected profit: +1 unit
• Standard deviation: ±11 units
• Actual results: anywhere from -21 to +23 units is normal

You need thousands of hands before edge dominates variance. Professional counters understand this viscerally. Amateurs often quit after a bad session, convinced "the count doesn't work."

The count works. You just have to survive long enough for the math to converge.

Conclusion: EV Is the Only Truth

Everything in advantage play reduces to EV:

• Basic strategy: the set of decisions that minimize negative EV
• Card counting: detecting when EV becomes positive
• Bet sizing: maximizing growth rate given positive EV and variance
• Session management: balancing EV of play against EV of camouflage

Casinos understand this. Their entire business model is ensuring every game has negative EV for players. Blackjack is unique because the EV shifts dynamically—and skilled players can identify and exploit the positive moments.

That's what advantage play really is: a relentless hunt for positive expected value.

See EV calculations in real-time for any hand situation. The calculator shows exact expected values for hit, stand, double, and split as deck composition changes.