Question

in this problem, round to four decimals when possible. 79% of all americans live in cities with population greater than 100,000 people. if 39 americans are randomly selected, find the probability that a. exactly 31 of them live in cities with population greater than 100,000 people. b. at most 31 of them live in cities with population greater than 100,000 people. c. at least 30 of them live in cities with population greater than 100,000 people. d. between 29 and 37 (including 29 and 37) of them live in cities with population greater than 100,000 people. hint: hint video on finding binomial probabilities +

Explanation:

This is a binomial probability problem where the binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, and $C(n,k)=\frac{n!}{k!(n - k)!}$, $n$ is the number of trials, $k$ is the number of successes, and $p$ is the probability of success on a single - trial. Here, $n = 39$, $p=0.79$, and $1 - p = 0.21$.

Step1: Calculate for part a (exactly 31 successes)

First, calculate the combination $C(39,31)=\frac{39!}{31!(39 - 31)!}=\frac{39!}{31!8!}=\frac{39\times38\times37\times36\times35\times34\times33\times32}{8\times7\times6\times5\times4\times3\times2\times1}=63574539$.
Then, $P(X = 31)=C(39,31)\times(0.79)^{31}\times(0.21)^{8}$
$P(X = 31)=63574539\times(0.79)^{31}\times(0.21)^{8}\approx0.1597$.

Step2: Calculate for part b (at most 31 successes)

$P(X\leq31)=\sum_{k = 0}^{31}C(39,k)\times(0.79)^{k}\times(0.21)^{39 - k}$.
We can use a binomial cumulative - distribution function. Using a calculator or software (e.g., TI - 84 Plus: binomcdf(39,0.79,31)), we get $P(X\leq31)\approx0.6777$.

Step3: Calculate for part c (at least 30 successes)

$P(X\geq30)=\sum_{k = 30}^{39}C(39,k)\times(0.79)^{k}\times(0.21)^{39 - k}=1-\sum_{k = 0}^{29}C(39,k)\times(0.79)^{k}\times(0.21)^{39 - k}$.
Using a binomial cumulative - distribution function (e.g., TI - 84 Plus: 1 - binomcdf(39,0.79,29)), we get $P(X\geq30)\approx0.6877$.

Step4: Calculate for part d (between 29 and 37 successes)

$P(29\leq X\leq37)=\sum_{k = 29}^{37}C(39,k)\times(0.79)^{k}\times(0.21)^{39 - k}=binomcdf(39,0.79,37)-binomcdf(39,0.79,28)$
Using a calculator or software, we get $P(29\leq X\leq37)\approx0.9597$.

Answer:

a. $0.1597$
b. $0.6777$
c. $0.6877$
d. $0.9597$