Calculating the Probability of Drawing a King or a Red Card from a Standard Deck

In the realm of probability, the concept of drawing a specific type of card from a well-shuffled deck of cards is a common scenario. One such scenario is determining the probability of drawing either a King or a Red Card. This article will walk you through the process using the principle of inclusion-exclusion.

The Problem and the Solution

The problem asks for the probability of drawing either a King or a Red Card from a standard deck of 52 playing cards. This is a classic application of the principle of inclusion-exclusion, which is a fundamental concept in probability theory.

Step 1: Count the Total Number of Favorable Outcomes

First, we need to count the total number of favorable outcomes for each event:

Number of Kings: There are 4 Kings in a deck (one for each suit). Number of Red Cards: There are 26 Red Cards (13 Hearts and 13 Diamonds). Number of Red Kings: There are 2 Red Kings (the King of Hearts and the King of Diamonds).

Step 2: Use the Inclusion-Exclusion Principle

The formula for the probability of either event A or event B occurring is given by:

$$P(A-B) P(A) P(B) - P(A cap B)$$

Where:

P(A) is the probability of drawing a King. P(B) is the probability of drawing a Red Card. P(A cap B) is the probability of drawing a Red King.

Step 3: Calculate Each Probability

Lets calculate the probabilities:

Probability of drawing a King (P(A)):

P(A) frac{4}{52}

Probability of drawing a Red Card (P(B)):

P(B) frac{26}{52}

Probability of drawing a Red King (P(A cap B)):

P(A cap B) frac{2}{52}

Step 4: Substitute into the Formula

Now, we substitute these values into the inclusion-exclusion formula:

$$P(A-B) frac{4}{52} frac{26}{52} - frac{2}{52}$$

Step 5: Simplify the Expression

Finally, we simplify the expression:

$$P(A-B) frac{4 26 - 2}{52} frac{28}{52} frac{7}{13}$$

The probability that the card drawn is either a King or a Red Card is frac{7}{13}.

Additional Considerations

For extra clarity, let's consider a few additional points:

If we asked for the probability of drawing an Ace, King, or a Red Card, the probabilities would be: Probability of drawing an Ace (P(A)):

P(A) frac{4}{52}

Probability of drawing a King (P(B)):

P(B) frac{4}{52}

Probability of drawing a Red Card (P(C)):

P(C) frac{26}{52}

Rewriting the formula for the combined event:

P(A-B-C) P(A) P(B) P(C) - P(A cap B) - P(B cap C) - P(A cap C) P(A cap B cap C)

Here, P(A cap B), P(B cap C), P(A cap C), P(A cap B cap C) are the probabilities of drawing two or three of specific cards at the same time.

Conclusion

The probability of drawing a King or a Red Card from a standard deck of 52 playing cards is calculated using the principle of inclusion-exclusion. The result is frac{7}{13}. Understanding and applying this principle is key to solving related probability problems.