Question

76% of all bald eagles survive their first year of life. give your answers as decimals, not percents. if 37 bald eagles are randomly selected, find the probability that

a. exactly 28 of them survive their first year of life.

b. at most 29 of them survive their first year of life.

c. at least 27 of them survive their first year of life.

d. between 25 and 32 (including 25 and 32) of them survive their first year of life.

Explanation:

This is a binomial probability problem. The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 37$, $p=0.76$, and $1 - p = 0.24$.

Step1: Calculate binomial coefficient for part a

For $k = 28$, calculate $C(37,28)=\frac{37!}{28!(37 - 28)!}=\frac{37!}{28!9!}=\frac{37\times36\times\cdots\times30}{9\times8\times\cdots\times1}=21860940$. Then $P(X = 28)=C(37,28)\times(0.76)^{28}\times(0.24)^{9}$.
$P(X = 28)=21860940\times(0.76)^{28}\times(0.24)^{9}\approx0.147$

Step2: Calculate cumulative probability for part b

$P(X\leq29)=\sum_{k = 0}^{29}C(37,k)\times(0.76)^{k}\times(0.24)^{37 - k}$. Using a binomial probability calculator or software (e.g., Excel's BINOM.DIST function), $P(X\leq29)\approx0.554$

Step3: Calculate cumulative probability for part c

$P(X\geq27)=1 - P(X\leq26)=\ 1-\sum_{k = 0}^{26}C(37,k)\times(0.76)^{k}\times(0.24)^{37 - k}\approx0.787$

Step4: Calculate cumulative probability for part d

$P(25\leq X\leq32)=\sum_{k = 25}^{32}C(37,k)\times(0.76)^{k}\times(0.24)^{37 - k}=P(X\leq32)-P(X\leq24)$. Using a binomial probability calculator, $P(25\leq X\leq32)\approx0.827$

Answer:

a. $0.147$
b. $0.554$
c. $0.787$
d. $0.827$