The probability that a randomly observed Radiata Pine tree will live for 15 years is 0.58.  This can be modelled using a Binomial distribution. 

If 6 Radiata Pine trees are randomly observed, calculate the probability that more than 1 of them will live for 15 years.

Round the answer to **3 decimal places.**

Let's solve the problem using the Binomial distribution formula.

### Step 1: Define the parameters.
- \( n = 6 \) (number of trials or trees)
- \( p = 0.58 \) (probability of success, i.e., a tree living for 15 years)
- The probability of failure \( q = 1 - p = 1 - 0.58 = 0.42 \).

We are asked to find the probability that **more than 1 tree** lives for 15 years. This means we need to find:

\[
P(X > 1)
\]

Where \( X \) follows a Binomial distribution: \( X \sim \text{Binomial}(n = 6, p = 0.58) \).

### Step 2: Find \( P(X > 1) \).

Instead of finding \( P(X > 1) \) directly, we can calculate its complement:

\[
P(X > 1) = 1 - P(X \leq 1)
\]

### Step 3: Calculate \( P(X \leq 1) \).

\[
P(X \leq 1) = P(X = 0) + P(X = 1)
\]

#### a) Calculate \( P(X = 0) \):

\[
P(X = 0) = \binom{6}{0} (0.58)^0 (0.42)^6 = (1) \cdot (0.42)^6
\]

\[
P(X = 0) = 0.42^6 \approx 0.005
\]

#### b) Calculate \( P(X = 1) \):

\[
P(X = 1) = \binom{6}{1} (0.58)^1 (0.42)^5 = 6 \cdot (0.58) \cdot (0.42)^5
\]

\[
P(X = 1) \approx 6 \cdot 0.58 \cdot 0.42^5 \approx 0.058
\]

Adding these together:

\[
P(X \leq 1) \approx 0.005 + 0.058 = 0.063
\]

### Step 4: Calculate \( P(X > 1) \).

\[
P(X > 1) = 1 - P(X \leq 1) = 1 - 0.063 = 0.937
\]

### Step 5: Round the answer to 3 decimal places.

\[
\boxed{0.937}
\]

Would you like to explore more about Binomial distributions or another aspect of this problem?

Here are 5 related questions:
1. What is the probability that exactly 3 trees will live for 15 years?
2. How would the probability change if 10 trees were observed instead of 6?
3. What is the expected number of trees to live for 15 years when observing 6 trees?
4. How do you calculate the variance of a Binomial distribution?
5. What is the probability that none of the trees will live for 15 years?

**Tip:** Remember, when dealing with probabilities involving “at least” or “more than” scenarios, it is often easier to find the complement.

The probability that a randomly observed Radiata Pine tree will live for 15 years is 0.58. This can be modelled using a Binomial distribution. If 6 Radiata Pine trees are randomly observed, calculate the probability that more than 1 of them will live for 15 years. Round the answer to 3 decimal places.

Learn how to calculate the probability that more than 1 out of 6 Radiata Pine trees will live for 15 years using the Binomial distribution. This problem involves complementary probability and binomial formulas, suitable for students in Grades 10-12.

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