Question 1199758
**a) Probability of exactly 17 repeat offenders:**

* This is a binomial probability problem.
* n = 35 (sample size)
* p = 0.49 (probability of success - repeat offender)
* q = 1 - p = 0.51 (probability of failure - not a repeat offender)
* x = 17 (number of successes)

* P(X = 17) = (35C17) * (0.49)^17 * (0.51)^(35-17) 
   * where (35C17) is the number of combinations of 35 things taken 17 at a time.

* Using a binomial probability calculator or statistical software:
   P(X = 17) ≈ 0.1223

**b) Probability of at most 17 repeat offenders:**

* P(X ≤ 17) = P(X = 0) + P(X = 1) + ... + P(X = 17)

* Use a cumulative binomial probability calculator or statistical software:
   P(X ≤ 17) ≈ 0.6143

**c) Probability of at least 16 repeat offenders:**

* P(X ≥ 16) = P(X = 16) + P(X = 17) + ... + P(X = 35)

* Alternatively:
   P(X ≥ 16) = 1 - P(X ≤ 15) 
   P(X ≥ 16) ≈ 1 - 0.4098 = 0.5902

**d) Probability of between 15 and 21 repeat offenders (inclusive):**

* P(15 ≤ X ≤ 21) = P(X = 15) + P(X = 16) + ... + P(X = 21)

* Use a cumulative binomial probability calculator or statistical software to find:
   P(15 ≤ X ≤ 21) ≈ 0.7586

**In summary:**

* a) P(X = 17) ≈ 0.1223
* b) P(X ≤ 17) ≈ 0.6143
* c) P(X ≥ 16) ≈ 0.5902
* d) P(15 ≤ X ≤ 21) ≈ 0.7586