Probability of Finding All Four Suits in a Deck of Playing Cards

When dealing with a standard deck of playing cards (52 cards), a natural question arises: what is the probability of drawing a hand where all four suits are represented? This article covers both the theoretical approach and the practical calculation of this probability.

Theoretical and Practical Approaches

Let's consider a full deck of 52 cards either shuffled or in its initial order. A common misconception might suggest that the probability can be less than 100%, but since the deck always contains all four suits, the probability is indeed 100%. However, when we look at the likelihood of drawing a hand with all four suits, we can explore the calculations in more detail.

Direct Calculation Method

The simplest method to approach this is to consider the probability of each card drawn. The first card can be from any suit, hence the probability is:

First draw: 52/52 1/1 Second draw: 39/51 Third draw: 26/50 Fourth draw: 13/49

You then multiply these probabilities together:

(1) 1 * (39/51) * (26/50) * (13/49) 6591/62475 ≈ 0.105498

Combinatorial Approach

Another method is to use combinatorics. We need to choose one card from each suit:

Choose 1 card from each of the 4 suits: 13C1

There are C(52, 4) ways to choose 4 cards from 52. Thus, the probability is:

C(13, 1)^4 / C(52, 4) 28561/270725 ≈ 0.105498

Permutation-Based Calculation

Another way to look at it is through permutations. Consider the arrangement where all four suits are different:

There are 4! (4 factorial) ways to arrange four different suits, which is 24. The probability of any one specific arrangement is: (1/4) * (13/51) * (13/50) * (13/49).

Thus, the total probability is:

24 * (1/4) * (13/51) * (13/50) * (13/49) 2197/20825 ≈ 0.105498

Replacing Cards vs. No Replacement

If cards are replaced after being drawn, the probability changes:

1/4 * 3/4 * 1/4 * 3/4 9/64 ≈ 0.140625

This shows that the probability of drawing all four suits without replacement is indeed lower than with replacement.

Conclusion

The probability of finding all four suits in a hand of 4 cards from a standard deck of 52 cards is approximately 0.105498. This result can be obtained through various calculations and can be verified through both combinatorial and direct probability methods.