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 the problem, we will find the probability that at least one of the five cards drawn is a face card. It's often easier to first calculate the probability of the complementary event (i.e., no face cards are drawn) and then subtract it from 1.

### Step 1: Determine the total number of face cards and non-face cards.
- There are 52 cards in a standard deck.
- There are 12 face cards (Jack, Queen, King of each suit).
- Thus, there are \(52 - 12 = 40\) non-face cards.

### Step 2: Calculate the probability of drawing a non-face card for each draw.
When the deck is shuffled and a card is drawn, the probability of not drawing a face card (i.e., drawing a non-face card) is \( \frac{40}{52} \).

### Step 3: Find the probability of all 5 drawn cards being non-face cards.
Since the card is replaced each time before drawing the next one, the probability remains constant for each draw:

\[
P(\text{no face cards}) = \left(\frac{40}{52}\right)^5 = \left(\frac{10}{13}\right)^5
\]

### Step 4: Calculate the complement of this probability.
The complement is the probability that at least one of the five cards drawn is a face card:

\[
P(\text{at least one face card}) = 1 - P(\text{no face cards}) = 1 - \left(\frac{10}{13}\right)^5
\]

### Step 5: Compute the numerical value.
\[
P(\text{at least one face card}) = 1 - \left(\frac{10}{13}\right)^5 \approx 1 - 0.4822 = 0.5178
\]

### Step 6: Round to one decimal place.
\[
P(\text{at least one face card}) \approx 0.5
\]

So, the probability that at least one of the five cards drawn is a face card is **0.5** or **50%**.

Would you like further details or have any questions about the process?

Here are 5 related questions you might find interesting:
1. What is the probability that exactly one of the five cards drawn is a face card?
2. What is the probability of drawing exactly two face cards out of the five draws?
3. How does the probability change if there is no replacement after each draw?
4. What is the probability of drawing a face card on the first draw?
5. How would the calculation differ if the deck had jokers included?

**Tip:** When dealing with probabilities, consider the complementary event to simplify calculations, especially when dealing with "at least one" type of scenarios.

Discover the probability of drawing at least one face card from a shuffled deck of playing cards. Learn step-by-step how to calculate probabilities for complementary events and independent draws. Round your answer to one decimal place.

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