ENGR 200 — First Midterm (Part I)

Name: _________________________ Total: ______ / 50

Part A — True/False (1.67 point each)

1. The probability of any event is always between 0 and 1, inclusive.
2. The probability of the sample space is always 1.
3. If two events are mutually exclusive, then P(A ∩ B) = P(A)P(B).
5. The complement of an event A contains all outcomes not in A.
6. For any event A, P(A) + P(A′) = 1.
7. If P(A)=0.3 and P(B)=0.5, then P(A ∪ B) must equal 0.8.
8. If A and B are disjoint, then P(A ∪ B) = P(A) + P(B).
9. If A ⊆ B, then P(A ∩ B) = P(B).
10. If A ⊆ B, then P(A ∪ B) = P(A).
11. Independence implies P(A ∩ B) = P(A)P(B).
12. A permutation is an arrangement of objects where order matters.
13. Choosing a president and a vice-president is a combination problem.
14. Choosing a group of 3 students out of 10 for a project is a permutation problem.
15. A combination is a selection of objects where order does not matter.
16. Choosing 3 toppings for a pizza (order irrelevant) is a combination problem.
17. Choosing the top 3 finishers in a race (order matters) is a combination problem.
18. Combinations count fewer possibilities than permutations for the same n and r.

Part B — Distributions Clarification (1.67 points each)

19. Inspect n=10 lightbulbs. Each fails independently with probability p=0.05. Find P(X=0), the probability that none are defective.
20. A coffee shop receives an average of 3 customers every 5 minutes. Find the probability that exactly 5 customers arrive in the next 5 minutes.
21. A password system is tested with n=8 attempts, each with success probability p=0.1, independently. What is the probability of exactly 2 successful logins?
22. A basketball player has success probability p=0.3 per shot. Find the probability the first basket occurs on the 4th shot.
23. Roll a fair die n=12 times. Find the probability of getting exactly 3 rolls that show “1.”
24. Flip a fair coin until you get the first head. Find the probability that it takes exactly 5 flips to get the first head.
25. A help desk receives 6 calls per hour on average. Find the probability of receiving at least 2 calls in the next 20 minutes.
26. The arrival time of an elevator is equally likely between 0 and 10 minutes. Find the probability it arrives within the first 3 minutes.
27. Roll a die n=50 times. Find the probability of getting at most 5 sixes.
28. Flip a coin with p=0.4 for heads until the first tail appears. Find the probability the first tail appears on the 3rd flip.
29. Earthquakes in a region follow a constant rate of 2 per month on average. Find the probability that exactly 3 earthquakes occur this month.
30. Check n=20 incoming emails. Each is spam with probability p=0.15, independently. Find the probability that exactly 4 emails are spam.
End of Part I Solutions