The manufacturer of a fertilizer guarantees that, with the aid of the fertilizer, 80% of planted seeds will germinate. Suppose the manufacturer is correct. If 6 seeds planted with the fertilizer are randomly selected, what is the probability that at least 5 of them germinate?

Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

To solve this problem, we can use the binomial probability formula. The situation described involves a binomial distribution because there are a fixed number of trials (6 seeds), two possible outcomes for each trial (a seed either germinates or it does not), and the probability of success (a seed germinating) is constant for each trial.

### Binomial Probability Formula

The probability of getting exactly \( k \) successes in \( n \) independent Bernoulli trials, with probability \( p \) of success on each trial, 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 \(\frac{n!}{k!(n-k)!}\),
- \( n \) is the total number of trials,
- \( k \) is the number of successful trials,
- \( p \) is the probability of success on each trial.

### Given Information

- \( n = 6 \) (number of seeds),
- \( p = 0.8 \) (probability that a seed germinates),
- We need to find the probability that at least 5 seeds germinate, i.e., \( P(X \geq 5) \).

### Calculating \( P(X \geq 5) \)

This is the sum of the probabilities of 5 seeds germinating and 6 seeds germinating:

\[
P(X \geq 5) = P(X = 5) + P(X = 6)
\]

#### 1. Calculate \( P(X = 5) \)

\[
P(X = 5) = \binom{6}{5} (0.8)^5 (0.2)^1
\]

Calculating each term:

- \(\binom{6}{5} = 6\)
- \((0.8)^5 = 0.32768\)
- \((0.2)^1 = 0.2\)

Substitute these values into the formula:

\[
P(X = 5) = 6 \times 0.32768 \times 0.2 = 0.393216
\]

#### 2. Calculate \( P(X = 6) \)

\[
P(X = 6) = \binom{6}{6} (0.8)^6 (0.2)^0
\]

Calculating each term:

- \(\binom{6}{6} = 1\)
- \((0.8)^6 = 0.262144\)
- \((0.2)^0 = 1\)

Substitute these values into the formula:

\[
P(X = 6) = 1 \times 0.262144 \times 1 = 0.262144
\]

#### 3. Sum the Probabilities

\[
P(X \geq 5) = P(X = 5) + P(X = 6) = 0.393216 + 0.262144 = 0.65536
\]

### Final Result

Rounded to two decimal places, the probability that at least 5 seeds germinate is **0.66**.

If you have any questions or need further explanation, feel free to ask!

### Follow-up Questions

1. How would the probability change if the germination rate increased to 90%?
2. What is the probability that exactly 4 seeds germinate?
3. How does the probability change if 10 seeds are planted instead of 6?
4. How do you calculate the mean and variance of a binomial distribution?
5. What is the cumulative distribution function (CDF) for a binomial distribution?
6. How can the normal approximation be used for a binomial distribution?
7. What if the probability of germination is unknown; how can it be estimated?
8. What other real-world scenarios can be modeled by a binomial distribution?

### Tip
When working with binomial probabilities, it's often helpful to use cumulative probability tables or technology (such as a calculator or software) to find probabilities for larger numbers of trials.

Calculate the probability that at least 5 out of 6 seeds germinate when using a fertilizer with an 80% germination rate. Learn about binomial probability and its application in real-world scenarios. Detailed step-by-step solution provided.

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