Cards are drawn with replacement from a deck till a spade is drawn. What is the probability that at least 4 draws are necessary?

To solve this problem, we can use the concept of geometric probability.

### Problem Statement Recap:
Cards are drawn with replacement from a standard deck of 52 cards until a spade is drawn. We are asked to find the probability that at least 4 draws are necessary to get a spade.

### Key Points to Consider:

1. **Total Number of Cards in the Deck:** 52
2. **Number of Spades in the Deck:** 13
3. **Probability of Drawing a Spade (in a single draw):**  
   \[
   P(\text{spade}) = \frac{13}{52} = \frac{1}{4}
   \]
4. **Probability of Not Drawing a Spade (in a single draw):**  
   \[
   P(\text{not spade}) = 1 - \frac{1}{4} = \frac{3}{4}
   \]

### Approach:

To find the probability that at least 4 draws are necessary to get a spade, it means that in the first 3 draws, we did not get a spade. Then, we might or might not get a spade in the 4th draw, but we only care about the probability that no spade was drawn in the first 3 draws.

**The probability that at least 4 draws are necessary to get a spade is equal to the probability that no spade is drawn in the first 3 draws.**

### Calculation:

The probability of not drawing a spade in a single draw is \(\frac{3}{4}\). Since the draws are independent (due to replacement), the probability of not drawing a spade in three consecutive draws is:

\[
\left(\frac{3}{4}\right)^3
\]

Calculating that gives:

\[
\left(\frac{3}{4}\right)^3 = \frac{3^3}{4^3} = \frac{27}{64}
\]

### Conclusion:

The probability that at least 4 draws are necessary to get a spade is \(\frac{27}{64}\).

Would you like more details or have any questions?

### Related Questions:

1. What is the probability that exactly 4 draws are necessary to get a spade?
2. How would the probability change if the deck had only 10 spades?
3. What is the expected number of draws needed to get a spade?
4. How does the probability change if the cards are drawn without replacement?
5. What is the probability that at most 3 draws are necessary to get a spade?

### Tip:

When dealing with probability problems involving repeated trials until a specific event occurs, geometric and binomial distributions often provide helpful frameworks.

Explore the probability of drawing at least 4 cards from a deck until a spade is drawn. Learn how to calculate geometric probabilities and use the concept of independent events in probability theory. Suitable for Grades 9-12.

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