Trees planted by a landscaping firm have a 90
% one-year survival rate, If they plant 16
 trees in a park, what is the following probabilities:

1. All the trees survive one year.
Answer: 
0.1351
2. At least 14
 trees survive one year.
Answer: 

We are dealing with a **binomial probability distribution** where:

- The probability of success \( p = 0.90 \) (since 90% of the trees survive).
- The number of trials \( n = 16 \) (since 16 trees are planted).
- The probability of failure \( q = 1 - p = 0.10 \).

The probability mass function for the binomial distribution is given by:

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

Where:
- \( X \) is the number of successes (trees surviving).
- \( k \) is the number of successful outcomes (trees surviving).

### Question 2: Probability that at least 14 trees survive

We need to find \( P(X \geq 14) \), which is the sum of the probabilities that 14, 15, or 16 trees survive.

\[
P(X \geq 14) = P(X = 14) + P(X = 15) + P(X = 16)
\]

Let’s compute this.The probability that at least 14 trees survive one year is approximately **0.7892**.

Would you like any further details or have any questions? Here are 5 related questions to expand your understanding:

1. How do we interpret the result of \( P(X \geq 14) \)?
2. What is the probability that fewer than 12 trees survive?
3. How would the probability change if the survival rate was 85% instead of 90%?
4. Can you calculate the expected number of trees that survive after one year?
5. How does the binomial distribution relate to real-world situations like this?

**Tip:** Binomial distribution is useful for modeling situations where you have a fixed number of independent trials, each with the same probability of success!

Trees planted by a landscaping firm have a 90% one-year survival rate. If they plant 16 trees in a park, what is the following probability: 1. All the trees survive one year. 2. At least 14 trees survive one year.

Learn how to calculate binomial probabilities for tree survival in a park. This problem explores a 90% survival rate for 16 trees and calculates the probability that all trees survive or at least 14 trees survive. Suitable for students in Grades 10-12.

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