The manufacturer of a fertilizer guarantees that, with the aid of the fertilizer, 65% of planted seeds will germinate. Suppose the manufacturer is correct. If 7 seeds planted with the fertilizer are randomly selected, what is the probability that at least 6 of Item germinate?

Carry your intermediate computations to at least 4 decimal places, and round your answer to 2 decimal places.

Let's calculate the probability that at least 6 out of 7 seeds will germinate using the binomial distribution formula, ensuring to carry intermediate calculations to at least four decimal places and rounding the final answer to two decimal places.

### Step 1: Identify the parameters
- \( n = 7 \): the number of trials (seeds).
- \( p = 0.65 \): the probability of success on each trial (germination).
- We want to find \( P(X \geq 6) \), where \( X \) is the number of seeds that germinate.

### Step 2: Define the binomial probability formula
The probability of exactly \( k \) successes in \( n \) independent trials is given by:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

### Step 3: Calculate \( P(X = 6) \) and \( P(X = 7) \)

1. **For \( k = 6 \):**

\[
P(X = 6) = \binom{7}{6} (0.65)^6 (0.35)^1
\]

\[
\binom{7}{6} = \frac{7!}{6!1!} = 7
\]

\[
P(X = 6) = 7 \times (0.65)^6 \times (0.35)
\]

\[
P(X = 6) = 7 \times 0.1160291 \times 0.35 = 7 \times 0.0406102 = 0.2843
\]

2. **For \( k = 7 \):**

\[
P(X = 7) = \binom{7}{7} (0.65)^7 \times (0.35)^0
\]

\[
\binom{7}{7} = 1
\]

\[
P(X = 7) = 1 \times (0.65)^7 = 0.0752
\]

### Step 4: Find the total probability \( P(X \geq 6) \)

\[
P(X \geq 6) = P(X = 6) + P(X = 7)
\]

\[
P(X \geq 6) = 0.2843 + 0.0752 = 0.3595
\]

### Final Answer:
The probability that at least 6 out of the 7 seeds will germinate is **0.36** (rounded to two decimal places).

---

Would you like more details or have any questions?

Here are 8 related questions you could explore:
1. What is the probability that exactly 5 seeds germinate?
2. How would the probability change if the success rate was 75%?
3. What is the probability that all 7 seeds germinate?
4. What is the probability that fewer than 3 seeds germinate?
5. How do you calculate the binomial coefficient for large numbers?
6. What is the expected number of seeds that will germinate?
7. Can this problem be solved using normal approximation?
8. How does the binomial distribution apply to real-world situations?

**Tip:** In binomial probability problems, always double-check intermediate values to ensure accuracy, especially when rounding.

Learn how to calculate the probability that at least 6 out of 7 seeds will germinate when using a fertilizer with a 65% germination rate. Includes step-by-step calculations and binomial distribution formula application. Suitable for Grades 10-12 students and anyone interested in probability theory.

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