Probability of Drawing Four Different Suits from a Deck with Replacement

In the realm of probability, the challenge of drawing four cards from a standard deck of 52 cards with replacement and ensuring each card comes from a different suit is an intriguing problem. This article delves into the mathematical principles and step-by-step process to determine this probability. By understanding the fundamental concepts and calculations involved, we can accurately predict the likelihood of such an event.

Understanding the Deck

A standard deck of playing cards consists of 52 cards, which are divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, ensuring a diverse range of possibilities for our probability calculation.

Calculating Total Outcomes

When drawing with replacement, the total number of outcomes for drawing 4 cards is determined by the simple formula:

524

This is because each card draw is an independent event, and each draw has 52 possible outcomes.

Calculating Favorable Outcomes

To achieve favorable outcomes where all four cards are from different suits, we must consider the following steps:

Choosing Suits

We need to select 4 suits from the 4 available suits, which is a straightforward process. There is only one way to do this since all suits are used:

C choose 4 suits 1

Arranging the Suits

The number of ways to arrange these 4 suits is calculated using the factorial of 4:

4! 4 × 3 × 2 × 1 24

Choosing Cards

For each of the 4 suits, we can draw any of the 13 cards from that suit. Hence, for each arrangement of suits, the number of ways to select the cards is:

134

Total Favorable Outcomes

The total number of favorable outcomes is the product of the ways to arrange the suits and the ways to choose the cards:

Total favorable outcomes 4! × 134

Substituting the values:

Total favorable outcomes 24 × 28561 685464

Calculating the Probability

The probability that all four cards are from different suits is given by the ratio of favorable outcomes to total outcomes:

P frac{4! × 13^4}{52^4}

Substituting the values:

P frac{24 × 28561}{7311616}

Calculating the numerator:

24 × 28561 685464

Simplifying the probability:

P frac{685464}{7311616}

Using the greatest common divisor (GCD), we simplify the fraction:

7311616 and 685464 GCD 8

P frac{685464 ÷ 8}{7311616 ÷ 8} frac{85683}{913952}

Therefore, the probability that all four cards drawn are from four different suits is:

P boxed{frac{85683}{913952}}