The Blackjack Decision Collector Problem

Introduction

How many hands of Blackjack would a player need to make every possible decision? This document explores this mathematical puzzle, applying the "coupon collector's problem" to Blackjack decisions.

Defining the Problem

In our analysis:

• We're considering all possible decision points a player might face in Blackjack

• We only care about card values (A, 2-9, 10) where 10 includes J, Q, K

• Suits don't matter

• Each decision point is defined by the player's cards and the dealer's upcard

Counting the Decision Points

How many distinct decision points exist?

In Blackjack, decision points are determined by the player's hand and the dealer's upcard:

• Hard Totals: 17 player totals (4-20)

• Soft Totals: 9 player totals (A,2 through A,10)

• Pairs: 9 different pairs (A,A through 9,9, with 10/J/Q/K treated as the same)

Total: (17 + 9 + 9) × 10 dealer upcards (2-10, A) = 350 distinct decision points

Probability Distribution of Decision Points

Using the probability distributions from Blackjack hand analysis data:

Decision Frequency Extremes

• Most common: Hard 12-16 vs Ten (approximately 1 in 50-75 hands)

• Rarest: Pair of same rank vs same rank dealer card (approximately 1 in 5,525 hands)

This extreme variation in probability (a factor of nearly 100x between most common and rarest) explains why collecting all decisions is so challenging.

Calculating the Required Hands

We can approach the problem of how many hands are needed using three increasingly realistic models:

Model 1: No Repetition (Idealized Scenario)

If we could somehow arrange to see each decision exactly once without any repeats (completely hypothetical) we would need exactly 350 hands.

This represents the absolute minimum number of hands required and serves as a theoretical lower bound.

Model 2: Equal Probabilities with Repetition

If we assume all 350 decisions occur with equal probability (1/350 each) and allow for natural repetition we would need at least 2,252 hands.

This is calculated using the standard coupon collector's formula: E[T] = N × H_N = 350 × 6.44 ≈ 2,252

Where:

• N = 350 distinct decisions

• H_N is the Nth harmonic number (1 + 1/2 + 1/3 + ... + 1/N) ≈ 6.44 for N=350

This equal-probability model already shows that repetition increases the required hands by a factor of about 6.4.

Model 3: Actual Probabilities with Repetition (Most Realistic)

When accounting for the true non-uniform probabilities of different decision points we would need at least 355,028 hands.

This dramatically higher number reflects the reality that:

• The most common decisions will be seen hundreds of times (for example, Hard 16 vs Dealer's 10 will occur roughly once every 74 hands, so approximately 4,800 times in 355,000 hands)

• The rarest decisions occur only about once every 5,525 hands (for example, Pair of Aces vs Dealer's Ace would be expected to occur only about 64 times in 355,000 hands)

• The expected time is dominated by waiting for the rarest decisions

The coupon collector's problem with non-uniform probabilities uses the formula: E[T] = ∑(i=1 to n) 1/p(i)

Where p(i) is the probability of seeing decision i in a single hand.

This third model demonstrates why the rarity of certain decisions makes complete observation exponentially more difficult than simpler models would suggest.

Conclusion

The mathematical analysis shows that seeing every possible Blackjack decision would require playing approximately 355,000 hands.

This demonstrates why complete observation of all possible decision points is practically impossible in real-world scenarios, and statistical inference from a smaller sample of common decisions is necessary for any practical analysis of playing strategy.