To find the probabilities, we can use the binomial probability formula:
P(x) = nCx * p^x * (1-p)^(n-x)
where:
P(x) is the probability of getting exactly x successes,
n is the total number of trials,
p is the probability of success in one trial, and
nCx represents the number of ways to choose x successes out of n trials.
Given:
p (probability of surviving the first year) = 75% = 0.75
q (probability of not surviving the first year) = 1 - p = 1 - 0.75 = 0.25
n (total number of bald eagles selected) = 35
a. Probability that exactly 28 of them survive their first year of life:
P(28) = 35C28 * (0.75)^28 * (0.25)^(35-28)
b. Probability that at most 25 of them survive their first year of life:
P(at most 25) = P(0) + P(1) + P(2) + ... + P(25)
Calculate the individual probabilities P(x) for x = 0 to 25, and then sum them up.
c. Probability that at least 25 of them survive their first year of life:
P(at least 25) = 1 - P(0) - P(1) - P(2) - ... - P(24)
Calculate the individual probabilities P(x) for x = 0 to 24, and then subtract the sum from 1.
d. Probability that between 25 and 29 (including 25 and 29) of them survive their first year of life:
P(25 to 29) = P(25) + P(26) + P(27) + P(28) + P(29)
Calculate the individual probabilities P(x) for x = 25 to 29, and then sum them up.
To get the answers, we need to calculate the values of P(x) for each case and add or subtract them accordingly. It involves some calculations, but it's straightforward once you have the probabilities for each value of x.