A Landscaping Firm Plants Trees Probability Problem

A landscaping firm plants 10 trees in a park, and the trees have a 95% one-year survival rate. What is the probability that at least 8 trees survive one year?

In this scenario, we have a landscaping firm that plants 10 trees in a park, and each tree has a 95% one-year survival rate. We want to determine the probability that at least 8 trees out of the planted 10 will survive after one year.

Final answer:

To calculate the probability of at least 8 trees surviving one year, the binomial probability formula should be used. Sum the probabilities of exactly 8, 9, and 10 trees surviving by using the formula where n is 10, p is 0.95, and k is each number of successful outcomes.

Explanation:

The landscaping firm plants 10 trees with each tree having a 95% survival rate, and we want to find the probability that at least 8 trees survive one year. Such a problem can be solved using the binomial probability formula, as each tree has only two possible outcomes (survives or doesn't survive), with the probability of survival being 0.95 for each tree.

Let's denote the random variable X as the number of trees surviving after one year. We are interested in P(X ≥ 8). This involves calculating probabilities for X equal to 8, 9, and 10. Using the binomial formula:

• P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

Where n is the number of trials, k is the number of successful outcomes, and p is the probability of a success on a given trial. Calculate P(X = 8), P(X = 9), and P(X = 10), and then sum these probabilities to find the total probability that at least 8 trees survive.

Without performing the actual calculations and since it's not an option in the multiple-choice answers, we cannot know the exact probability, but the correct approach to solving the problem is using the binomial probability formula as described.