Cards are drawn without replacement from a deck of 52 cards until a spade appears. What is the probability that at least 4 draws are necessary?

To solve the problem, we need to find the probability that at least 4 cards are drawn before a spade appears in a standard deck of 52 cards.

### Key Points:
- A standard deck has 52 cards, including 13 spades.
- We are looking for the probability that the first spade appears on the 4th draw or later.

### Steps to Solve:

1. **Calculate the Probability for the First 3 Cards**:
   - For the first three draws to not be spades, we must only draw non-spade cards.

   The number of non-spade cards in the deck is \(52 - 13 = 39\).

   For the first draw, the probability of drawing a non-spade is:
   \[
   \frac{39}{52}
   \]

   For the second draw, assuming the first card was not a spade, there are 51 cards left, of which 38 are non-spades:
   \[
   \frac{38}{51}
   \]

   For the third draw, assuming the first two cards were not spades, there are 50 cards left, of which 37 are non-spades:
   \[
   \frac{37}{50}
   \]

   So, the probability that the first 3 cards drawn are not spades is:
   \[
   \frac{39}{52} \times \frac{38}{51} \times \frac{37}{50}
   \]

2. **Calculate the Required Probability**:
   - The required probability is the probability that all the first 3 cards are non-spades, because this means that the first spade appears on the 4th draw or later.
   \[
   P(\text{at least 4 draws}) = \frac{39}{52} \times \frac{38}{51} \times \frac{37}{50}
   \]

3. **Final Computation**:
   Calculate the numerical value of the expression:
   \[
   P(\text{at least 4 draws}) = \frac{39}{52} \times \frac{38}{51} \times \frac{37}{50} \approx 0.409
   \]

### Answer:
The probability that at least 4 draws are necessary to get the first spade is approximately **0.409**.

Would you like more details on any part of the solution, or do you have any further questions?

Here are 5 related questions to consider:
1. What is the probability that the first spade appears on the 1st draw?
2. How does the probability change if the deck had only 40 cards with 10 spades?
3. What is the probability that exactly 3 cards are drawn before the first spade appears?
4. How would the probability change if the deck were shuffled and replaced after each draw?
5. What is the expected number of draws before the first spade appears?

**Tip:** In probability problems involving drawing without replacement, the total number of possible outcomes decreases with each draw, making these problems ideal for conditional probability approaches.

Explore the probability of drawing cards without replacement from a deck of 52 cards until a spade appears. Learn how to calculate the probability that at least 4 draws are necessary with step-by-step explanations and examples. Suitable for advanced high school students and anyone interested in probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation