Practice Problems: Probability Distributions
Question 1
A quality control engineer at a manufacturing plant monitors the production of bolts. Historical data shows that 3% of the bolts produced are defective. In a random sample of 200 bolts, what is the expected number of defective bolts?
(a) 3 bolts
(b) 6 bolts
(c) 9 bolts
(d) 12 bolts
Question 2
A civil engineer is analyzing traffic flow at an intersection. Vehicles arrive at an average rate of 4 vehicles per minute following a Poisson distribution. What is the probability that exactly 3 vehicles will arrive in a given minute?
(a) 0.156
(b) 0.195
(c) 0.238
(d) 0.287
Question 3
An electrical engineer tests circuit boards where the time to failure follows an exponential distribution with a mean time of 5000 hours. What is the probability that a circuit board will fail before 2000 hours of operation?
(a) 0.247
(b) 0.330
(c) 0.451
(d) 0.632
Question 4
A mechanical engineer is inspecting welds on a pipeline. The probability of finding a defective weld is 0.05. If the engineer inspects 10 welds, what is the probability of finding exactly 2 defective welds?
(a) 0.0746
(b) 0.0988
(c) 0.1142
(d) 0.1386
Question 5
A structural engineer measures the compressive strength of concrete samples. The strength follows a normal distribution with a mean of 4000 psi and a standard deviation of 400 psi. What is the probability that a randomly selected sample has a strength greater than 4600 psi?
(a) 0.067
(b) 0.093
(c) 0.134
(d) 0.159
Question 6
An industrial engineer monitors a manufacturing process where defects occur randomly at a rate of 2.5 defects per hour. Assuming a Poisson distribution, what is the probability of observing exactly 5 defects in a 2-hour period?
(a) 0.146
(b) 0.176
(c) 0.205
(d) 0.234
Question 7
A reliability engineer analyzes component failures where the probability of failure is constant at 0.02 per component. In a system with 50 independent components, what is the variance of the number of failed components?
(a) 0.49
(b) 0.68
(c) 0.98
(d) 1.22
Question 8
A transportation engineer studies vehicle speeds on a highway. Speeds are normally distributed with mean 65 mph and standard deviation 8 mph. What percentage of vehicles travel between 55 mph and 75 mph?
(a) 68.3%
(b) 78.9%
(c) 84.1%
(d) 89.4%
Question 9
A chemical engineer monitors a reactor where accidents occur at an average rate of 0.8 per year following a Poisson distribution. What is the probability that no accidents will occur in a given year?
(a) 0.335
(b) 0.410
(c) 0.449
(d) 0.527
Question 10
An environmental engineer measures pollutant concentration levels that are normally distributed with mean 50 μg/m³ and standard deviation 12 μg/m³. What is the 90th percentile concentration level?
(a) 62.4 μg/m³
(b) 65.4 μg/m³
(c) 68.2 μg/m³
(d) 71.6 μg/m³
Question 11
A systems engineer evaluates a communication network where message arrivals follow a Poisson process at 6 messages per minute. What is the expected time (in seconds) until the next message arrives?
(a) 5 seconds
(b) 10 seconds
(c) 15 seconds
(d) 20 seconds
Question 12
A manufacturing engineer conducts quality control where parts are randomly selected for inspection. If the process has a 15% defect rate and 8 parts are inspected, what is the probability of finding at least one defective part?
(a) 0.657
(b) 0.728
(c) 0.796
(d) 0.843
Question 13
A geotechnical engineer measures soil bearing capacity values that follow a normal distribution with mean 2500 psf and variance 90,000 psf². What is the standard deviation of the bearing capacity?
(a) 225 psf
(b) 250 psf
(c) 275 psf
(d) 300 psf
Question 14
A power systems engineer analyzes outages at a substation that occur at an average rate of 1.5 outages per month following a Poisson distribution. What is the probability of exactly 2 outages occurring in a given month?
(a) 0.209
(b) 0.251
(c) 0.298
(d) 0.334
Question 15
A water resources engineer measures daily water demand that is normally distributed with mean 2.4 million gallons and standard deviation 0.6 million gallons. What is the probability that demand exceeds 3.0 million gallons on a given day?
(a) 0.136
(b) 0.159
(c) 0.184
(d) 0.227
Question 16
An aerospace engineer tests components where the time to failure follows an exponential distribution with mean 8000 flight hours. What is the probability that a component survives beyond 10,000 flight hours?
(a) 0.217
(b) 0.287
(c) 0.356
(d) 0.422
Question 17
A construction project manager tracks equipment breakdowns that occur according to a Poisson process at 3 breakdowns per week. If the project runs for 2 weeks, what is the standard deviation of the number of breakdowns?
(a) 1.73
(b) 2.00
(c) 2.45
(d) 3.00
Question 18
A quality engineer samples parts from a production line where 8% are defective. In a sample of 100 parts, what is the probability of finding fewer than 5 defective parts?
(a) 0.176
(b) 0.221
(c) 0.265
(d) 0.312
Question 19
A network engineer monitors server requests that arrive at a rate of 10 per second following a Poisson distribution. What is the probability that more than 12 requests arrive in a given second?
(a) 0.242
(b) 0.283
(c) 0.324
(d) 0.365
Question 20
A materials engineer measures the tensile strength of steel specimens that are normally distributed with mean 72 ksi and standard deviation 6 ksi. If specifications require strength between 65 ksi and 80 ksi, what percentage of specimens meet the specifications?
(a) 81.9%
(b) 85.7%
(c) 89.3%
(d) 92.8%
