Please show your work (draw figures such as Venn diagrams and write equations whenever possible) before checking the answer and explanations. The questions in the exams will be similar to these in the practice exercises.

Problem 1.   The Coffee Shop: At a coffee shop, 60% of customers order coffee, 40% order tea, and 25% order both coffee and tea. What is the probability that a randomly selected customer orders coffee or tea?

A) 0.85
B) 0.75
C) 1.00
D) 0.65

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Problem 2.   The Exam Results In a class, 70% of students passed Math, 80% passed English, and 60% passed both. What is the probability that a randomly chosen student passed at least one of the two subjects?

A) 0.90
B) 0.94
C) 0.85
D) 1.50

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Problem 3.   The Library Survey A library survey found that 30% of members borrow fiction books, 50% borrow non-fiction, and 20% borrow both. What percentage borrow only fiction?

A) 10%
B) 20%
C) 30%
D) 50%

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Problem 4.   The Flight Delays At an airport, the probability that Flight A is delayed is 0.2, and the probability that Flight B is delayed is 0.15. If the events are independent, what is the probability that both flights are delayed?

A) 0.35
B) 0.03
C) 0.30
D) 0.05

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Problem 5.   The Weather Forecast The probability of rain on Saturday is 0.4, and the probability of rain on Sunday is 0.3. If the probability that it rains on at least one day is 0.55, are the events “rain on Saturday” and “rain on Sunday” mutually exclusive?

A) Yes
B) No
C) Not enough information
D) They are independent

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Problem 6.   The Job Applicants Out of 100 applicants, 65 have a degree, 45 have experience, and 30 have both. Given that an applicant has a degree, what is the probability they also have experience?

A) 0.30
B) 0.461
C) 0.667
D) 0.45

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Problem 7.   The Soccer Team A soccer team wins with probability 0.6, draws with probability 0.25, and loses with probability 0.15. What is the probability they do not lose?

A) 0.85
B) 0.75
C) 0.40
D) 0.60

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Problem 8.   The Medical Test A medical test for a disease has a false positive rate of 5%. If 1 in 1000 people have the disease, and a random person tests positive, which probability concept is needed to find the chance they actually have the disease?

A) Addition rule
B) Multiplication rule for independent events
C) Conditional probability
D) Mutually exclusive rule

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Problem 9.   The Restaurant Orders At a restaurant, the probability a customer orders dessert is 0.3. The probability they order coffee is 0.5. If ordering dessert and ordering coffee are mutually exclusive, what is the probability a customer orders dessert or coffee?

A) 0.15
B) 0.80
C) 0.65
D) 0.80

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Problem 10.   The Traffic Lights You pass through three independent traffic lights on your commute. Each has a 0.4 chance of being red. What is the probability all three are red?

A) 1.2
B) 0.064
C) 0.4
D) 0.12

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Problem 11.  You roll a fair six-sided die. This is an example of:

A) A continuous uniform random variable
B) A discrete uniform random variable
C) A normal random variable
D) A binomial random variable

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Problem 12.   A game show wheel is divided into 10 equal-sized wedges numbered 1 through 10. When spun fairly, the number where it stops follows:

A) A continuous uniform distribution
B) A discrete non-uniform distribution
C) A discrete uniform distribution
D) A Poisson distribution

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Problem 13.   Probability Density Function For a continuous uniform random variable X on [a, b], the height of its probability density function between a and b is:

A) 1
B) (b - a)
C) 1/(b - a)
D) 0.5

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Problem 14.   Random Time Appointment You schedule a meeting randomly between 2:00 PM and 3:00 PM. The arrival time X (in minutes after 2:00) follows a continuous uniform distribution. What is P(X > 45)?

A) 0.25
B) 0.50
C) 0.75
D) 0.85

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Problem 15.   You roll a fair six-sided die. Let X be the outcome. What is P(2 ≤ X ≤ 4)?

A) 1/6
B) 1/3
C) 1/2
D) 2/3

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Problem 16.   Mixed Probability Question X follows a continuous uniform distribution on [0, 10]. What is P(3 ≤ X ≤ 7)?

A) 0.3
B) 0.4
C) 0.5
D) 0.7

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Problem 17.   The Gym Members A gym finds that 40% of members use the treadmill, 50% use weights, and 30% use both. Are using the treadmill and using weights independent?

A) Yes, because P(T) × P(W) = 0.4 × 0.5 = 0.2, which is not equal to P(T and W) = 0.3
B) No, because P(T) × P(W) ≠ P(T and W)
C) Yes, because they can both happen
D) No, because they are mutually exclusive

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Problem 18.   The Assembly Line A factory has two machines, A and B. The probability machine A fails is 0.1, and the probability machine B fails is 0.05. If failures are independent, what is the probability both are working?

A) 0.95
B) 0.855
C) 0.15
D) 0.995

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Problem 19.   The Magazine Readers A survey finds that 40% read Newsweek, 30% read Time, and 20% read both. What percentage read exactly one of the two magazines?

A) 20% B) 30% C) 50% D) 70%

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Problem 20.   The Car Insurance Data shows 10% of drivers have an accident in a year, 20% get a speeding ticket, and 3% have both. What is P(accident | speeding ticket)?

A) 0.03
B) 0.15
C) 0.30
D) 0.50

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