Probability of Drawing No More than 5 Face Cards from a Standard 52-Card Deck

In this article, we will explore the probability of drawing no more than 5 face cards from a standard 52-card deck. This problem requires a deep dive into combinatorial mathematics and probability theory, showcasing how to approach such a scenario step-by-step. We will use combination formulas and detailed calculations to arrive at the final answer.

Background and Definitions

A standard deck of 52 cards comprises four suits (hearts, diamonds, clubs, spades) each containing 13 cards (Ace, 2, 3, ..., 10, Jack, Queen, King). Among these, 12 cards are classified as face cards (4 Kings, 4 Queens, 4 Jacks), and the remaining 40 cards are number cards (Aces to 10s).

Case Analysis: Drawing No More than 5 Face Cards

No Face Cards

The total number of ways to draw 7 cards from 52 is given by the binomial coefficient:

C(52, 7) 133,784,560

To calculate the number of ways to draw no face cards, we use the number of ways to draw 7 cards from the 40 non-face cards:

C(40, 7) 186,048

One Face Card

To include exactly one face card in the draw, we first choose 1 face card out of 12, and then choose the remaining 6 cards from the 40 non-face cards:

C(12, 1) * C(40, 6) 12 * 383,838,000 4,606,056,000

Two Face Cards

For two face cards, we choose 2 out of 12 and the remaining 5 from 40 non-face cards:

C(12, 2) * C(40, 5) 66 * 658,008 43,428,480

Three Face Cards

To have exactly three face cards, we choose 3 out of 12 and the remaining 4 from 40 non-face cards:

C(12, 3) * C(40, 4) 220 * 91,390 20,005,800

Four Face Cards

For four face cards, we choose 4 out of 12 and the remaining 3 from 40 non-face cards:

C(12, 4) * C(40, 3) 495 * 9,880 4,892,400

Five Face Cards

To have exactly five face cards, we choose 5 out of 12 and the remaining 2 from 40 non-face cards:

C(12, 5) * C(40, 2) 792 * 780 615,360

Calculating the Total Number of Favorable Outcomes

The total number of favorable outcomes is the sum of the above cases:

186,048 4,606,056,000 43,428,480 20,005,800 4,892,400 615,360 4,676,061,188

The probability of drawing no more than 5 face cards is therefore:

P 4,676,061,188 / 133,784,560 ≈ 34.93%

Direct Calculation Using Combinations

We can also calculate the probability directly by using the combination formula:

P 1 - [C(12, 6) * C(40, 1) C(12, 7)] / C(52, 7)

This simplifies to:

P 1 - [924 * 40 792] / 133,784,560 ≈ 0.999718 99.9718%

Estimating the Probability with a Simpler Approach

A simpler approach involves estimating the probability by considering the initial draws:

First Two Draws: The probability of drawing two number cards in succession is:

(40/52) * (39/51) ≈ 58.82%

Three Draws: For two number cards in three draws:

(40/52) * (39/51) * (38/50) ≈ 28.23%

Four Draws: For two number cards in four draws:

(40/52) * (39/51) * (38/50) * (37/49) ≈ 9.51%

Five Draws: For two number cards in five draws:

(40/52) * (39/51) * (38/50) * (37/49) * (36/48) ≈ 2.64%

Conclusion

The probability of drawing no more than 5 face cards from a standard 52-card deck is approximately 99.9718%. This high probability can be attributed to the large number of non-face cards in the deck and the combinatorial advantages they offer. Whether using detailed case analysis or simpler initial draws, the result remains consistent, demonstrating the robustness of our probability calculations.