a card from a pack of 52 playing cards is lost. From the remaining cards of the pack, three cards are drawn at random (without replacement) and are found to be all clubs. Find the probability of the lost card being a club.

To solve this problem, we can apply **Bayes' theorem**. We are interested in finding the probability that the lost card is a club given that three drawn cards are all clubs. Let's define the events:

- \( C \) is the event that the lost card is a club.
- \( D \) is the event that three drawn cards are all clubs.

We want to find \( P(C | D) \), the probability that the lost card is a club given that three cards drawn are all clubs.

### Step 1: Bayes' Theorem

Bayes' theorem gives us the following formula:

\[
P(C | D) = \frac{P(D | C) \cdot P(C)}{P(D)}
\]

Where:
- \( P(C | D) \) is the probability that the lost card is a club, given that three drawn cards are clubs.
- \( P(D | C) \) is the probability of drawing three clubs, given that the lost card is a club.
- \( P(C) \) is the prior probability that the lost card is a club.
- \( P(D) \) is the total probability of drawing three clubs, regardless of which card was lost.

### Step 2: Assign Values to the Probabilities

#### 1. Prior Probability: \( P(C) \)
Since the pack has 52 cards, of which 13 are clubs, the probability of the lost card being a club is:

\[
P(C) = \frac{13}{52} = \frac{1}{4}
\]

#### 2. Probability of Drawing Three Clubs Given That the Lost Card Is a Club: \( P(D | C) \)
If the lost card is a club, there are 12 clubs left in the deck of 51 cards. So the probability of drawing three clubs from these 51 cards is:

\[
P(D | C) = \frac{\binom{12}{3}}{\binom{51}{3}} = \frac{\frac{12 \times 11 \times 10}{3 \times 2 \times 1}}{\frac{51 \times 50 \times 49}{3 \times 2 \times 1}} = \frac{220}{117600}
\]

#### 3. Probability of Drawing Three Clubs Given That the Lost Card Is Not a Club: \( P(D | C^c) \)
If the lost card is not a club, then there are still 13 clubs in the deck of 51 cards. The probability of drawing three clubs in this case is:

\[
P(D | C^c) = \frac{\binom{13}{3}}{\binom{51}{3}} = \frac{\frac{13 \times 12 \times 11}{3 \times 2 \times 1}}{\frac{51 \times 50 \times 49}{3 \times 2 \times 1}} = \frac{286}{117600}
\]

#### 4. Total Probability of Drawing Three Clubs: \( P(D) \)
The total probability of drawing three clubs can be found using the law of total probability:

\[
P(D) = P(D | C) \cdot P(C) + P(D | C^c) \cdot P(C^c)
\]

Substitute the values we found:

\[
P(D) = \left( \frac{220}{117600} \times \frac{1}{4} \right) + \left( \frac{286}{117600} \times \frac{3}{4} \right)
\]

\[
P(D) = \frac{220}{470400} + \frac{858}{470400} = \frac{1078}{470400}
\]

### Step 3: Apply Bayes' Theorem

Now we can substitute all the values into Bayes' theorem:

\[
P(C | D) = \frac{P(D | C) \cdot P(C)}{P(D)} = \frac{\frac{220}{117600} \times \frac{1}{4}}{\frac{1078}{470400}}
\]

Simplify the expression:

\[
P(C | D) = \frac{\frac{220}{470400}}{\frac{1078}{470400}} = \frac{220}{1078} \approx 0.204
\]

Thus, the probability that the lost card is a club, given that the three drawn cards are clubs, is approximately **0.204** or **20.4%**.

### Summary:
The probability that the lost card is a club is approximately 0.204, or 20.4%.

Would you like further details or clarification? Here are some related questions you could explore:

1. How does the probability change if two cards were drawn instead of three?
2. What if the deck had a different number of clubs, such as 10 or 15?
3. What is the probability if we know that the drawn cards are not all clubs, but a different suit?
4. How does Bayesian probability apply to other card-related problems?
5. What are the real-world applications of Bayes' theorem?

**Tip:** Bayesian probability is a great tool for situations where you have prior knowledge (like the probability of a lost card being a club) and want to update that knowledge based on new evidence (like the cards drawn).

A card from a pack of 52 playing cards is lost. From the remaining cards of the pack, three cards are drawn at random (without replacement) and are found to be all clubs. Find the probability of the lost card being a club.

Explore how Bayes' Theorem can be applied to determine the probability that the lost card is a club, given that three drawn cards from a deck are clubs. The solution involves combinatorics and probability theory, including key calculations and step-by-step guidance. Suitable for high school students in Grades 11-12.

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