chapter 4: discrete probability and the binomial distribution score: 15/18 answered: 7/8 question 8 60% of…

chapter 4: discrete probability and the binomial distribution score: 15/18 answered: 7/8 question 8 60% of all violent felons in the prison system are repeat offenders. if 48 violent felons are randomly selected, find the probability that a. exactly 28 of them are repeat offenders. b. at most 28 of them are repeat offenders. c. at least 30 of them are repeat offenders. d. between 25 and 31 (including 25 and 31) of them are repeat offenders. hint: hint video on finding binomial probabilities +

chapter 4: discrete probability and the binomial distribution score: 15/18 answered: 7/8 question 8 60% of all violent felons in the prison system are repeat offenders. if 48 violent felons are randomly selected, find the probability that a. exactly 28 of them are repeat offenders. b. at most 28 of them are repeat offenders. c. at least 30 of them are repeat offenders. d. between 25 and 31 (including 25 and 31) of them are repeat offenders. hint: hint video on finding binomial probabilities +

Answer

Explanation:

Step1: Identify binomial parameters

Let $n = 48$ (number of trials/sample size), $p=0.6$ (probability of success, i.e., a felon is a repeat - offender), and $q = 1 - p=0.4$. The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times q^{n - k}$, where $C(n,k)=\frac{n!}{k!(n - k)!}$.

Step2: Calculate probability for part a

For $k = 28$: $C(48,28)=\frac{48!}{28!(48 - 28)!}=\frac{48!}{28!20!}$ $P(X = 28)=C(48,28)\times(0.6)^{28}\times(0.4)^{20}$ Using a calculator or software for combinations and powers: $C(48,28)\approx1.34\times10^{13}$ $(0.6)^{28}\approx4.88\times10^{-7}$ $(0.4)^{20}\approx1.09951\times10^{-8}$ $P(X = 28)\approx1.34\times10^{13}\times4.88\times10^{-7}\times1.09951\times10^{-8}\approx0.072$

Step3: Calculate probability for part b

$P(X\leq28)=\sum_{k = 0}^{28}C(48,k)\times(0.6)^{k}\times(0.4)^{48 - k}$. Using a binomial - probability calculator or software (e.g., in R: pbinom(28,48,0.6)), we get $P(X\leq28)\approx0.382$

Step4: Calculate probability for part c

$P(X\geq30)=1 - P(X\lt30)=1-\sum_{k = 0}^{29}C(48,k)\times(0.6)^{k}\times(0.4)^{48 - k}$. Using a binomial - probability calculator or software (e.g., in R: 1 - pbinom(29,48,0.6)), we get $P(X\geq30)\approx0.544$

Step5: Calculate probability for part d

$P(25\leq X\leq31)=\sum_{k = 25}^{31}C(48,k)\times(0.6)^{k}\times(0.4)^{48 - k}$. Using a binomial - probability calculator or software (e.g., in R: pbinom(31,48,0.6)-pbinom(24,48,0.6)), we get $P(25\leq X\leq31)\approx0.719$

Answer:

a. $0.072$ b. $0.382$ c. $0.544$ d. $0.719$