A magician shuffles a standard deck of playing cards and allows an audience member to pull out a card, look at it, and replace it in the deck. Four additional people do the same. Find the probability that of the 5 cards drawn, at least 1 is a face card. (Round your answer to one decimal place.)

To solve this problem, we'll calculate the probability that **at least one of the 5 cards drawn** is a face card. A face card in a standard deck of playing cards is a Jack, Queen, or King. There are 3 face cards in each of the 4 suits, making a total of:

\[
3 \times 4 = 12 \text{ face cards}
\]

The total number of cards in a standard deck is 52.

### Step 1: Calculate the probability of the complementary event

The complementary event to "at least one of the 5 cards is a face card" is "none of the 5 cards is a face card." We will first calculate this complementary probability and then subtract it from 1.

1. **Number of non-face cards in the deck:**

\[
52 - 12 = 40 \text{ non-face cards}
\]

2. **Probability that a single drawn card is not a face card:**

\[
\frac{40}{52} = \frac{10}{13}
\]

3. **Probability that all 5 cards drawn are not face cards:**

Since the audience member replaces the card after each draw, each draw is independent. Thus, the probability that all 5 cards drawn are not face cards is:

\[
\left( \frac{10}{13} \right)^5
\]

### Step 2: Calculate the complementary probability

Now let's compute this value:

\[
\left( \frac{10}{13} \right)^5 \approx (0.7692)^5
\]

Calculating:

\[
(0.7692)^5 \approx 0.247
\]

### Step 3: Calculate the probability of at least one face card

The probability that at least one of the 5 cards drawn is a face card is the complement of the above probability:

\[
1 - \left( \frac{10}{13} \right)^5 \approx 1 - 0.247 = 0.753
\]

### Step 4: Round to one decimal place

Rounding to one decimal place, the probability is:

\[
\boxed{0.8}
\]

So, the probability that at least one of the 5 cards drawn is a face card is **0.8 (or 80%)**. 

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What is the probability that exactly two of the five cards are face cards?
2. How would the probability change if only 3 cards were drawn instead of 5?
3. What is the probability that all five drawn cards are face cards?
4. How does the probability change if the deck is not shuffled after each draw?
5. How would this problem change if there were jokers in the deck?

**Tip:** When solving probability problems, consider whether events are independent (replacement) or dependent (without replacement).

Discover how to calculate the probability that at least one of five cards drawn from a deck is a face card. This solution explores concepts of probability, combinatorics, and complementary events, using a standard deck of playing cards. Ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation