Question

chapter 4: discrete probability and the binomial distribution
score: 6/18 answered: 4/8
question 5
61% of all americans are home owners. if 42 americans are randomly selected, find the probability that
a. exactly 25 of them are are home owners.
b. at most 25 of them are are home owners.
c. at least 23 of them are home owners.
d. between 21 and 27 (including 21 and 27) of them are home owners.
hint:
video on finding binomial probabilities +

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, $n = 42$, $p=0.61$, and $1 - p = 0.39$.

Step1: Calculate combination for part a

For $k = 25$, calculate $C(42,25)=\frac{42!}{25!(42 - 25)!}=\frac{42!}{25!17!}$.
Using a calculator or software, $C(42,25)\approx1.44\times10^{11}$. Then $P(X = 25)=C(42,25)\times(0.61)^{25}\times(0.39)^{17}$.
$P(X = 25)\approx1.44\times10^{11}\times1.19\times10^{- 6}\times1.17\times10^{- 9}\approx0.162$.

Step2: Calculate cumulative probability for part b

To find $P(X\leq25)$, we sum up $P(X = k)$ for $k$ from $0$ to $25$. Using a binomial probability calculator or software (e.g., in R: pbinom(25,42,0.61)), $P(X\leq25)\approx0.377$.

Step3: Calculate complementary cumulative probability for part c

$P(X\geq23)=1 - P(X\leq22)$. Using a binomial probability calculator (e.g., in R: 1 - pbinom(22,42,0.61)), $P(X\geq23)\approx0.897$.

Step4: Calculate cumulative probability difference for part d

$P(21\leq X\leq27)=P(X\leq27)-P(X\leq20)$. Using a binomial probability calculator (e.g., in R: pbinom(27,42,0.61)-pbinom(20,42,0.61)), $P(21\leq X\leq27)\approx0.704$.

Answer:

a. $0.162$
b. $0.377$
c. $0.897$
d. $0.704$