Question 1168408: 64% 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 30 of them are repeat offenders.
c. At least 32 of them are repeat offenders.
d. Between 28 and 34 (including 28 and 34) of them are repeat offenders.
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem using the binomial probability distribution.
**Given:**
* Probability of a violent felon being a repeat offender (p) = 64% = 0.64
* Number of violent felons selected (n) = 48
**a) Exactly 28 of them are repeat offenders.**
We want to find P(X = 28), where X is the number of repeat offenders.
The binomial probability formula is:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
* nCk = n! / (k! * (n-k)!)
* n = 48
* k = 28
* p = 0.64
P(X = 28) = (48C28) * (0.64)^28 * (0.36)^20
* 48C28 = 48! / (28! * 20!) ≈ 1.8396 * 10^13
* (0.64)^28 ≈ 0.00000216
* (0.36)^20 ≈ 1.3407 * 10^-9
P(X = 28) = 1.8396 * 10^13 * 0.00000216 * 1.3407 * 10^-9
P(X = 28) ≈ 0.0534
**b) At most 30 of them are repeat offenders.**
We want to find P(X ≤ 30), which is the sum of P(X = k) for k = 0 to 30.
P(X ≤ 30) = Σ [ (48Ck) * (0.64)^k * (0.36)^(48-k) ] for k = 0 to 30
Using a calculator or software:
P(X ≤ 30) ≈ 0.686
**c) At least 32 of them are repeat offenders.**
We want to find P(X ≥ 32), which is the sum of P(X = k) for k = 32 to 48.
P(X ≥ 32) = Σ [ (48Ck) * (0.64)^k * (0.36)^(48-k) ] for k = 32 to 48
Using a calculator or software:
P(X ≥ 32) ≈ 0.231
**d) Between 28 and 34 (including 28 and 34) of them are repeat offenders.**
We want to find P(28 ≤ X ≤ 34), which is the sum of P(X = k) for k = 28 to 34.
P(28 ≤ X ≤ 34) = Σ [ (48Ck) * (0.64)^k * (0.36)^(48-k) ] for k = 28 to 34
Using a calculator or software:
P(28 ≤ X ≤ 34) ≈ 0.669
**Answers:**
a) Approximately 0.0534
b) Approximately 0.686
c) Approximately 0.231
d) Approximately 0.669
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
64% 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 30 of them are repeat offenders.
c. At least 32 of them are repeat offenders.
d. Between 28 and 34 (including 28 and 34) of them are repeat offenders.
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As it regularly happens with @CPhill' solutions on Binomial distribution,
all calculations (a), (b), (c) and (d) in the post by @CPhill are incorrect.
Below are my correct answers.
(a) P(X=28) = 0.0837
(b) P(X <= 30) = 0.4681
(c) P(X >= 32) = 0.4126
(d) P(28 <= X <= 34) = P(X <= 34) - P(X <= 27) = 0.8732 - 0.1662 = 0.707.
To check your/my calculations, use online calculator
https://stattrek.com/online-calculator/binomial
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