Question

66% of all bald eagles survive their first year of life. If 35 bald eagles are randomly selected, find the probability that

a. Exactly 23 of them survive their first year of life.
b. At most 22 of them survive their first year of life.
c. At least 23 of them survive their first year of life.
d. Between 21 and 26 (including 21 and 26) of them survive their first year of life.

Answer 1

To calculate the probabilities, we can use the binomial probability formula. The probability of exactly 23 surviving is approximately 0.1365. The probability of at most 22 surviving is approximately 0.0037. The probability of at least 23 surviving is approximately 0.9963. The probability of between 21 and 26 surviving is approximately 0.9633.

Step-by-step explanation:

To find the probability in each case, we can use the binomial probability formula. The formula is:

() = ()(^)(^(−))

Where:

() is the probability of getting exactly successes,

is the number of trials,

is the probability of success on each trial, and

is the probability of failure on each trial.

Let's calculate each probability:

a. Exactly 23 surviving:

(=23) = (3523)(0.66^23)(0.34^12) ≈ 0.1365

b. At most 22 surviving:

(≤22) = (=0) + (=1) + ... + (=22)

(≤22) = (350)(0.66^0)(0.34^35) + (351)(0.66^1)(0.34^34) + ... + (3522)(0.66^22)(0.34^13) ≈ 0.0037

c. At least 23 surviving:

(≥23) = 1 - (<23)

(≥23) = 1 - (≤22) ≈ 0.9963

d. Between 21 and 26 surviving:

(21≤≤26) = (=21) + ... + (=26)

(21≤≤26) = (3521)(0.66^21)(0.34^14) + (3522)(0.66^22)(0.34^13) + ... + (3526)(0.66^26)(0.34^9) ≈ 0.9633