Probability of Drawing Two Cards from Different Suits

In this article, we will explore the probability of drawing two cards from a standard deck of 52 cards such that each card comes from a different suit. We will break down the logical steps to determine this probability and explain the calculations involved.

Introduction to Probability and Combinations

Probability is a measure of the likelihood that a given event will occur. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Calculating the Total Number of Ways to Draw 2 Cards

Let's start by finding the total number of ways to draw 2 cards from a deck of 52 cards. The number of combinations can be determined using the combination formula:

Total Ways to Draw 2 Cards

n binom{52}{2} frac{52 times 51}{2} 1326

This means there are 1326 different ways to draw 2 cards from a deck of 52 cards.

Ways to Draw 2 Cards from Different Suits

To determine the number of ways to draw two cards from different suits, we break down the problem:

Options for the First Card

The first card can be of any suit, so there are 52 different options for the first card.

Options for the Second Card

The second card must be from a different suit. Since there are 4 suits, the first card takes one of these, leaving 3 suits. Therefore, each of these suits has 13 cards, giving us:

3 times 13 39

options for the second card.

Total Number of Ways to Draw 2 Cards from Different Suits

52 times 39 2028

Therefore, there are 2028 ways to draw two cards from different suits.

Probability Calculation

The probability of drawing two cards from different suits is given by the ratio of the number of favorable outcomes to the total outcomes:

P_{text{different suits}} frac{2028}{1326}

This fraction simplifies to:

P_{text{different suits}} frac{2028 div 6}{1326 div 6} frac{338}{221} approx 1.528

However, this indicates a miscalculation in the counting of favorable outcomes. Let's recalculate the probability correctly:

Correct Probability Calculation

The number of ways to draw 2 cards from the same suit is:

There are 4 suits, and for each suit, there are

binom{13}{2} 78

ways to choose 2 cards. Thus, the total ways to draw 2 cards from the same suit is:

4 times 78 312

The probability that both cards are from the same suit is:

P_{text{same suit}} frac{312}{1326}

Now, we find the probability that the two cards are from different suits:

P_{text{different suits}} 1 - P_{text{same suit}} 1 - frac{312}{1326} frac{1326 - 312}{1326} frac{1014}{1326}

Simplifying this fraction gives:

P_{text{different suits}} frac{507}{663} approx 0.764

Thus, the probability that the two drawn cards are from different suits is approximately 0.764 or 76.4%.