Researchers found that 30% of the larch trees in a certain geographical area had been damaged by fungi. (No fun, guy.) Consider a sample of 20 (independent) beach trees from the area.

(a) What is the probability that fewer than half are damaged by fungi?

(b) What is the probability that more than 14 are damaged?

(d) Find the 75th percentile for the number of damaged trees in groups of 20.

Used binomial distribution formula but I have no clue if it's even close.

a0 7

To solve these problems, we can use the binomial distribution formula. The binomial distribution is used when there are only two possible outcomes for each trial, and the probability of success remains constant.

In this case, we have a sample of 20 trees and the probability of a tree being damaged by fungi is 30%.

The binomial distribution formula is:
P(X = k) = (n choose k) * p^k * q^(n-k)

Where:
P(X = k) is the probability of getting exactly k successes,
n is the number of trials,
p is the probability of success,
q is the probability of failure (1 - p), and
(n choose k) is the binomial coefficient, which is calculated as n! / (k! * (n-k)!)

Now let's solve the given questions.

(a) What is the probability that fewer than half are damaged by fungi?
To calculate this probability, we need to sum up the probabilities of having 0, 1, 2, ..., 9, or 10 damaged trees out of 20.
P(X < 10) = P(X = 0) + P(X = 1) + ... + P(X = 9)

(b) What is the probability that more than 14 are damaged?
To calculate this probability, we need to sum up the probabilities of having 15, 16, ..., 20 damaged trees out of 20.
P(X > 14) = P(X = 15) + P(X = 16) + ... + P(X = 20)

(c) Find the 75th percentile for the number of damaged trees in groups of 20.
The 75th percentile represents the value for which 75% of the data falls below it. To find it, we need to calculate the cumulative probability and find the value of k for which P(X ≤ k) ≥ 0.75.

Now that we know the steps, we can proceed with calculating these probabilities using the binomial distribution formula.