in this problem, round to four decimals when possible. 79% of all americans live in cities with population…

Explanation:

This is a binomial probability problem where the binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, $P(X\leq k)=\sum_{i = 0}^{k}C(n,i)\times p^{i}\times(1 - p)^{n - i}$, $P(X\geq k)=1 - P(X<k)$ and $P(a\leq X\leq b)=\sum_{i = a}^{b}C(n,i)\times p^{i}\times(1 - p)^{n - i}$, with $n$ being the number of trials, $k$ the number of successes, $p$ the probability of success on a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 39$, $p=0.79$, and $1 - p = 0.21$.

Step1: Calculate binomial coefficient for part a

For $k = 31$, $C(39,31)=\frac{39!}{31!(39 - 31)!}=\frac{39!}{31!8!}=\frac{39\times38\times37\times36\times35\times34\times33\times32}{8\times7\times6\times5\times4\times3\times2\times1}=63574539$. $P(X = 31)=C(39,31)\times(0.79)^{31}\times(0.21)^{8}$ $P(X = 31)=63574539\times(0.79)^{31}\times(0.21)^{8}\approx0.1499$.

Step2: Calculate cumulative probability for part b

We need to find $P(X\leq31)=\sum_{i = 0}^{31}C(39,i)\times(0.79)^{i}\times(0.21)^{39 - i}$. Using a binomial probability calculator or software (e.g., in R: pbinom(31,39,0.79)), $P(X\leq31)\approx0.5886$.

Step3: Calculate probability for part c

$P(X\geq30)=1 - P(X<30)=1-\sum_{i = 0}^{29}C(39,i)\times(0.79)^{i}\times(0.21)^{39 - i}$. Using a binomial probability calculator or software (e.g., in R: 1 - pbinom(29,39,0.79)), $P(X\geq30)\approx0.7873$.

Step4: Calculate probability for part d

$P(29\leq X\leq37)=\sum_{i = 29}^{37}C(39,i)\times(0.79)^{i}\times(0.21)^{39 - i}$. Using a binomial probability calculator or software (e.g., in R: pbinom(37,39,0.79)-pbinom(28,39,0.79)), $P(29\leq X\leq37)\approx0.9585$.

Answer:

a. $0.1499$ b. $0.5886$ c. $0.7873$ d. $0.9585$