Solved Problems

A batch of concrete cylinders has a $10\%$ chance ($p=0.10$) of failing a strength test. If $5$ cylinders are randomly selected for testing ($n=5$), what is the probability that exactly $1$ fails?

Vehicles arrive at a toll booth at an average rate of $\lambda = 3$ per minute. What is the probability of exactly $2$ arrivals in a given minute?

The number of potholes on a highway follows a Poisson distribution with an average of $2$ potholes per mile ($\lambda = 2$). What is the probability that there is at least $1$ pothole in a specific mile?

A robotic welder on a manufacturing line produces a defective joint with a probability of $p = 0.05$. The machine produces joints until a defective one is found, at which point it is shut down for calibration. What is the probability that the machine produces exactly $10$ good joints before the first defective one?

A construction company orders a batch of $n = 8$ steel beams. The probability that a beam has a minor surface defect is $p = 0.15$. What is the probability that at most $1$ beam has a surface defect?

A traffic engineer observes that an intersection experiences an average of $4$ red-light running incidents per day. What is the probability of observing exactly $7$ incidents over a $2$-day period?

In a soil sampling process, the probability of finding a specific contaminant in a given core sample is $p = 0.20$. Samples are tested sequentially. What is the probability that the first contaminated sample is found after the $3$rd test?

A civil engineer inspects a fleet of $50$ construction vehicles. The probability that any given vehicle requires immediate maintenance is $p = 0.08$. What is the expected number of vehicles requiring maintenance, and what is the standard deviation?