Probability of Drawing Four Spades from a Deck of Cards: A Comprehensive Guide

Calculating the probability of drawing four spades from a standard deck of 52 cards is a fascinating application of combinatorial probability. This article provides a detailed step-by-step guide on how to determine this probability, supported by calculations and simulations. We will cover the fundamental concepts, the mathematical approach, and a practical simulation to validate our solution.

Understanding the Basics

A standard deck of cards contains 52 cards, divided into four suits: spades, hearts, diamonds, and clubs. Each suit has 13 cards. The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.

Theoretical Calculation

To find the probability of drawing four spades from a deck of 52 cards, we need to use the concept of combinations. The combination formula is defined as:

[ binom{n}{k} frac{n!}{k!(n-k)!} ]

Step 1: Determine the Total Number of Ways to Choose 4 Cards from 52

The total number of ways to choose 4 cards from a deck of 52 is given by:

[ binom{52}{4} frac{52!}{4!(52-4)!} frac{52!}{4! cdot 48!} frac{52 cdot 51 cdot 50 cdot 49}{4 cdot 3 cdot 2 cdot 1} 270725 ]

Step 2: Determine the Number of Ways to Choose 4 Spades from 13

The number of ways to choose 4 spades from the 13 available spades is:

[ binom{13}{4} frac{13!}{4!(13-4)!} frac{13!}{4! cdot 9!} frac{13 cdot 12 cdot 11 cdot 10}{4 cdot 3 cdot 2 cdot 1} 715 ]

Step 3: Calculate the Probability

The probability that all four cards drawn are spades is given by the ratio of the number of favorable outcomes to the total number of outcomes:

[ P(text{all 4 are spades}) frac{binom{13}{4}}{binom{52}{4}} frac{715}{270725} approx 0.00264 ]

Validation through Simulation

To validate our theoretical result, we can perform a simulation. In this case, we will run 1000 simulations of drawing 4 cards from a deck of 52 cards, record the number of times we get 4 spades, and use the average to approximate the probability.

The R code provided for the simulation is as follows:

simulations  1000
create a deck define spades as 1 everything else as 0
deck  rep(1, 13) rep(0, 52 - 13)  # Deck of 52 cards with 13 spades
spades  rep(NA, simulations)
for (i in 1:simulations) {
  hand  sample(deck, 4, replace  FALSE) # Draw 4 cards without replacement
  spades[i]  sum(hand) # Count the number of spades in the hand
}

mean(spades)  # Should get 1

Running this simulation, we observe that the average number of times 4 spades are drawn (out of 1000 simulations) is approximately 1.012.

Alternative Calculation Method

Another way to calculate the probability is by considering the individual probabilities of drawing a spade at each step:

[ left( frac{13}{52} right) times left( frac{12}{51} right) times left( frac{11}{50} right) times left( frac{10}{49} right) frac{13 times 12 times 11 times 10}{52 times 51 times 50 times 49} frac{17160}{6497400} approx 0.00264 ]

Conclusion

The probability that all four cards dealt are spades is approximately 0.00264 or 0.264%. This result is consistent with both our theoretical calculation and the simulation.

In summary, the probability of drawing four spades from a deck of 52 cards is a clear demonstration of combinatorial probability principles and highlights the power of both theoretical analysis and empirical validation.