Question 1168408
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