Introduction to the Probability of Drawing 5 Red Cards

Are you looking to understand the probabilities involved in drawing cards from a standard deck? Calculating the probability of drawing 5 red cards is a classic problem in combinatorics and probability theory. This guide will walk you through the steps and formulas you need to determine the chances of such an event.

Understanding the Deck

A standard deck of playing cards comprises 52 cards, divided into two colors: red and black. Each color has 26 cards, with 13 hearts and 13 diamonds in the red cards category. Our aim is to calculate the probability of drawing 5 red cards without replacement.

Step 1: Total Ways to Draw 5 Cards from the Deck

The total number of ways to draw 5 cards from a 52-card deck can be calculated using the combination formula. The combination formula, denoted as C(n, k), is given by:

$C(n, k) frac{n!}{k!(n-k)!}$

Applying this formula to our scenario:

$C(52, 5) frac{52!}{5!(52-5)!} frac{52 times 51 times 50 times 49 times 48}{5 times 4 times 3 times 2 times 1} 2598960$

Step 2: Ways to Draw 5 Red Cards

The total number of ways to draw 5 red cards from the 26 available red cards follows the same combination formula:

$C(26, 5) frac{26!}{5!(26-5)!} frac{26 times 25 times 24 times 23 times 22}{5 times 4 times 3 times 2 times 1} 65780$

Step 3: Calculating the Probability

The probability of drawing 5 red cards is the ratio of the number of ways to draw 5 red cards to the total number of ways to draw 5 cards:

$P frac{C(26, 5)}{C(52, 5)} frac{65780}{2598960} approx 0.0253$

This means the probability of drawing 5 red cards is approximately 0.0253, or 2.53%.

Alternative Method: Direct Calculation Using Multiplication Principle

An alternative method to calculate the probability involves multiplying the individual probabilities of drawing each red card consecutively without replacement:

$P_{5R} frac{26}{52} times frac{25}{51} times frac{24}{50} times frac{23}{49} times frac{22}{48} frac{253}{9996} approx 2.5%$

Conclusion

By applying the principles of combinatorics and probability, we have calculated the probability of drawing 5 red cards from a standard deck of 52 cards. Whether you use the combination formula or the direct calculation method, the result confirms that the chances of achieving this specific outcome are quite low, just 2.53%.