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 four decimal places, and round your answer to two decimal places.

This problem involves finding the probability that at least 6 out of 7 seeds will germinate, given that each seed has a 65% chance of germination. This situation can be modeled using the binomial distribution.

### 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 the binomial probability formula:

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

Where:
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).

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

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

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

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

\[
P(X = 7) = \binom{7}{7} (0.65)^7 (1-0.65)^{0} 
\]

### Step 4: Compute the probabilities

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

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

\[
P(X = 6) = 7 \times (0.11602906) \times 0.35 = 0.2838
\]

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

\[
P(X = 7) = \frac{7!}{7!0!} (0.65)^7 = 1 \times (0.07530352) = 0.0753
\]

### Step 5: Find the total probability \( P(X \geq 6) \)

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

\[
P(X \geq 6) = 0.2838 + 0.0753 = 0.3591
\]

### Final Answer:
The probability that at least 6 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 70%?
3. What is the probability that fewer than 4 seeds germinate?
4. How do you calculate the binomial coefficient for large numbers?
5. Can this problem be solved using normal approximation?
6. How does the binomial distribution apply to real-world situations?
7. What is the expected number of seeds that will germinate?
8. How does increasing the number of seeds affect the distribution?

**Tip:** When working with binomial probabilities, it's helpful to use a calculator or software that handles binomial functions to avoid manual errors, especially with larger numbers.

Explore how to calculate the probability that at least 6 out of 7 seeds will germinate using the binomial distribution. Understand the impact of seed germination rates and fertilizer effectiveness with detailed step-by-step computations. Suitable for Grades 10-12.

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