Probabilities of Drawing Cards from the Same Suit: Analytical and Recursive Approaches

The question of finding the probability that three randomly drawn cards from a deck of 52 playing cards are of the same suit is a classic problem in probability theory and has numerous applications in games like poker. This article explores both analytical and recursive methods to solve this problem, providing a comprehensive understanding through detailed calculations and explanations.

Introduction to the Problem

Consider a scenario where three cards are randomly drawn from a full deck of 52 playing cards. What is the probability that these three cards are all of the same suit? A deck of 52 cards consists of four suits (hearts, diamonds, clubs, and spades), with each suit containing 13 cards. The problem can be solved using both recursive and analytical methods, each offering insights into the probability calculation.

Recursive Method

Let's start with the recursive method, which involves breaking down the problem into simpler steps and solving each step progressively.

Probability Calculation for the First Two Cards

When the first card is drawn, there are 52 cards, and any suit can be chosen. The probability that the second card drawn is of the same suit as the first is given by:

[ frac{12}{51} ]

This is because 12 out of the remaining 51 cards are of the same suit.

Probability Calculation for the Third Card

Now, we need to calculate the probability that the third card is also of the same suit. Given that the first two cards were of the same suit, there are now 11 cards remaining in that suit from the original 51 cards. Therefore, the probability is:

[ frac{11}{50} ]

The joint probability of all three cards being of the same suit is the product of these probabilities:

[ frac{12}{51} times frac{11}{50} frac{132}{2550} 0.05176470588235294 ]

Analytical Method

For the analytical method, we use combinatorial mathematics to calculate the probability directly. Let's break down the calculations step by step.

Total Number of Ways to Draw 3 Cards

The total number of ways to draw 3 cards from 52 is given by the binomial coefficient:

[ binom{52}{3} frac{52!}{3!(52-3)!} frac{52 times 51 times 50}{3 times 2 times 1} 22100 ]

Number of Ways to Draw 3 Cards from the Same Suit

The number of ways to choose 3 cards from a single suit of 13 cards is:

[ binom{13}{3} frac{13!}{3!(13-3)!} frac{13 times 12 times 11}{3 times 2 times 1} 286 ]

Since there are 4 suits, the total number of ways to draw 3 cards from the same suit is:

[ 4 times 286 1144 ]

Calculating the Probability

The probability that all three cards are of the same suit is the ratio of the favorable outcomes to the total outcomes:

[ frac{1144}{22100} frac{1312114}{525150} 0.05176470588 ]

This confirms the earlier recursive calculation.

General Non-Recursive Solution with Analytic Combinatorics

For a more general solution that does not rely on recursion, we can use the principle of multiplicative counting. The probability of drawing a card of the same suit as the first in subsequent draws is calculated as follows:

[ prod_{i1}^{2} frac{13-i}{53-i} ]

This can be simplified to:

[ frac{12}{52} times frac{11}{51} times frac{10}{50} frac{1320}{132600} 0.009918032786885245 times 4 0.039672131147541085 approx 0.0397 ]

This approach can be generalized for drawing more than three cards from the same suit, yielding accurate probabilities as:

[ prod_{i1}^{n-1} frac{13-i}{53-i} text{ for } n5 text{ cards} approx 0.001980792 ]

This value represents the probability of a five-card flush in poker, aligning well with the initial recursive calculation.

Conclusion

Both the recursive and analytical methods provide accurate probabilities for drawing cards from the same suit. The recursive method breaks down the problem into simpler parts, while the analytical method uses combinatorial mathematics for a direct calculation. The general non-recursive solution with analytic combinatorics offers a flexible and powerful tool for solving such problems in probability theory.

References

[1] Probability Theory. Wikipedia. [2] Combinatorics: Combinations and Permutations. Math is Fun. [3] Combination. Wikipedia.