ENGR 200 — First Midterm

Name: _________________________ Date: __________ Total: ______/100

Instructions

Show clear work for all problems. Give answers as simplified fractions or decimals rounded to four places unless stated otherwise. No formulas or hints are provided on the exam.

Part A (Chapter 2) — Bayes’ Theorem: Dorm Problems

1) Three Dorms (A/B/C).

The campus has Dorm A, Dorm B, and Dorm C. The fractions of students in each dorm are: Dorm A 25%, Dorm B 25%, Dorm C 30%. The probability a randomly chosen student is female within each dorm is: Dorm A 60%, Dorm B 45%, Dorm C 35%. Given that a randomly chosen student is female, compute the probability she is in each dorm: P(Dorm A | Female), P(Dorm B | Female), and P(Dorm C | Female).

2) Four Dorms (A/B/C/D).

Dorm populations: A 25%, B 25%, C 30%, D 20%. The probability a student is female within each dorm: A 60%, B 45%, C 35%, D 50%. Given that a student is female, find the probability she is in each dorm: P(A | Female), P(B | Female), P(C | Female), P(D | Female).

3) Five Dorms (A/B/C/D/E).

Dorm populations: A 15%, B 20%, C 25%, D 20%, E 20%. The probability a student is female within each dorm: A 70%, B 60%, C 50%, D 40%, E 30%. Given that a student is female, find the probability she is in each dorm: P(A | Female), P(B | Female), P(C | Female), P(D | Female), P(E | Female).

Part B — Permutations & Combinations

4) Unordered selection of dorms.

Choose 3 dorms out of 5 (A–E) for a pilot program. Order does not matter. How many different sets of 3 dorms can be chosen? Give an integer.

5) Ordered schedule of dorm visits.

You will visit 3 distinct dorms out of 5 on Day 1, Day 2, and Day 3. Order matters. How many different 3-day schedules are possible? Give an integer.

Part C (Chapter 3) — Poisson, Binomial, and Geometric

6) Poisson — No typos across pages.

A proofreader finds on average 0.8 typos per page. For 3 pages, what is the probability of no typos in total?

7) Poisson — At least one call.

A help desk receives 6 calls per hour on average. What is the probability of receiving at least one call in the next 10 minutes?

8) Binomial — Exactly four successes.

A device succeeds with probability p = 0.30 each trial, independently, for n = 10 trials. Find P(X = 4).

9) Binomial — At most one defect.

Each item is defective with probability p = 0.05, independently. For n = 20 items, find P(X ≤ 1).

10) Geometric — First success on the 5th trial.

Independent trials with success probability p = 0.20. What is P(X = 5), i.e., the probability the first success occurs on trial 5?

End of Exam