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.
    Show solution 
    True.  Axioms: 0 ≤ P(A) ≤ 1.
 
  
    2. The probability of the sample space is always 1.
    Show solution 
    True.  By definition P(S)=1.
 
  
    3. If two events are mutually exclusive, then P(A ∩ B) = P(A)P(B).
    Show solution 
    False.  If disjoint, P(A∩B)=0; P(A)P(B) generally ≠ 0.
 
  
    5. The complement of an event A contains all outcomes not in A.
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    6. For any event A, P(A) + P(A′) = 1.
    Show solution 
    True.  Complements partition S.
 
  
    7. If P(A)=0.3 and P(B)=0.5, then P(A ∪ B) must equal 0.8.
    Show solution 
    False.  P(A∪B)=P(A)+P(B)−P(A∩B).
 
  
    8. If A and B are disjoint, then P(A ∪ B) = P(A) + P(B).
    Show solution 
    
   
  
    9. If A ⊆ B, then P(A ∩ B) = P(B).
    Show solution 
    False.  If A⊆B then A∩B=A ⇒ P(A∩B)=P(A).
 
  
    10. If A ⊆ B, then P(A ∪ B) = P(A).
    Show solution 
    False.  Then A∪B=B ⇒ P(A∪B)=P(B).
 
  
    11. Independence implies P(A ∩ B) = P(A)P(B).
    Show solution 
    
   
  
    12. A permutation is an arrangement of objects where order matters.
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    13. Choosing a president and a vice-president is a combination problem.
    Show solution 
    False.  Roles differ → permutation.
 
  
    14. Choosing a group of 3 students out of 10 for a project is a permutation problem.
    Show solution 
    False.  Order doesn’t matter → combination.
 
  
    15. A combination is a selection of objects where order does not matter.
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    16. Choosing 3 toppings for a pizza (order irrelevant) is a combination problem.
    Show solution 
    
   
  
    17. Choosing the top 3 finishers in a race (order matters) is a combination problem.
    Show solution 
    False.  Order matters → permutation.
 
  
    18. Combinations count fewer possibilities than permutations for the same n and r.
    Show solution 
    
   
  
  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.
    Show solution 
    
      Model: Binomial . P(X=0)=C(10,0)(0.05)^0(0.95)^{10}=(0.95)^{10}.
      P ≈ 0.5987 . Excel: =BINOM.DIST(0,10,0.05,FALSE) 
     
   
  
    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.
    Show solution 
    
      Model: Poisson  with λ=3 per 5 minutes. P(X=5)=e^{-3}3^5/5!
      P ≈ 0.1008 . Excel: =POISSON.DIST(5,3,FALSE) 
     
   
  
    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?
    Show solution 
    
      Model: Binomial . P(X=2)=C(8,2)(0.1)^2(0.9)^6
      P ≈ 0.1488 . Excel: =BINOM.DIST(2,8,0.1,FALSE) 
     
   
  
    22. A basketball player has success probability p=0.3 per shot. Find the probability the first basket occurs on the 4th shot.
    Show solution 
    
      Model: Geometric  (trials to first success). P(X=4)=(1−p)^{3}p.
      P ≈ 0.1029 . Excel: =POWER(1-0.3,3)*0.3 
     
   
  
    23. Roll a fair die n=12 times. Find the probability of getting exactly 3 rolls that show “1.”
    Show solution 
    
      Model: Binomial , n=12, p=1/6. P(X=3)=C(12,3)(1/6)^3(5/6)^9
      P ≈ 0.1974 . Excel: =BINOM.DIST(3,12,1/6,FALSE) 
     
   
  
    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.
    Show solution 
    
      Model: Geometric  with p=0.5. P(X=5)=(1−0.5)^{4}·0.5
      P = 0.03125 . Excel: =POWER(0.5,4)*0.5 
     
   
  
    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.
    Show solution 
    
      Model: Poisson  with λ=6×(20/60)=2. P(X≥2)=1−[P(0)+P(1)]=1−e^{−2}(1+2).
      P ≈ 0.5940 . Excel: =1-POISSON.DIST(1,2,TRUE) 
     
   
  
    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.
    Show solution 
    
      Model: Uniform (0,10). P(0≤X≤3)=(3−0)/(10−0)=3/10.
      P = 0.3000 . Excel: =3/10 
     
   
  
    27. Roll a die n=50 times. Find the probability of getting at most 5 sixes.
    Show solution 
    
      Model: Binomial , n=50, p=1/6. P(X≤5)=∑k=0..5  C(50,k)(1/6)^k(5/6)^{50−k}
      P ≈ 0.1388 . Excel: =BINOM.DIST(5,50,1/6,TRUE) 
     
   
  
    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.
    Show solution 
    
      Model: Geometric  for “first tail” with success prob q=0.6. Need HH then T: (0.4)^2(0.6).
      P = 0.0960 . Excel: =POWER(0.4,2)*0.6 
     
   
  
    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.
    Show solution 
    
      Model: Poisson  with λ=2. P(X=3)=e^{−2}·2^3/3!
      P ≈ 0.1804 . Excel: =POISSON.DIST(3,2,FALSE) 
     
   
  
    30. Check n=20 incoming emails. Each is spam with probability p=0.15, independently. Find the probability that exactly 4 emails are spam.
    Show solution 
    
      Model: Binomial . P(X=4)=C(20,4)(0.15)^4(0.85)^{16}
      P ≈ 0.1821 . Excel: =BINOM.DIST(4,20,0.15,FALSE)