Solved Problems

A geotechnical engineer is classifying a soil sample based on its primary constituent (Gravel, Sand, Silt, or Clay) and its moisture condition (Dry or Wet). What is the sample space for this classification?

A construction project is monitored for two independent risks: weather delays (Event $W$) and material shortages (Event $M$). Describe the events $W \cup M$ and $W \cap M$ in practical terms.

An intern assigns probabilities to three mutually exclusive and exhaustive outcomes of a compressive strength test: $P(\text{Low}) = 0.20$, $P(\text{Medium}) = 0.55$, and $P(\text{High}) = 0.30$. Are these assigned probabilities valid according to the axioms of probability?

In structural design, a steel column can fail by yielding (Event $Y$) or by buckling (Event $B$). If yielding and buckling cannot happen simultaneously on the exact same cross-section at the same time, what is the relationship between these two events? Are they independent?

A new highway segment requires the selection of a subbase material, a base course material, and a surface course type. There are $3$ options for the subbase, $4$ options for the base course, and $2$ options for the surface course. How many unique pavement structures can be designed?

A construction project involves $5$ critical tasks that must be completed one after the other in sequence. In how many different orders can these $5$ tasks be scheduled?

From a batch of $12$ concrete cylinders, a quality control engineer must randomly select $4$ cylinders to be tested for compressive strength at $28$ days. In how many ways can this selection be made?

A surveyor is setting up signal flags along a route. They have $8$ flags in total: $3$ identical red flags, $3$ identical blue flags, and $2$ identical yellow flags. How many distinct sequences of flags can be arranged in a line?

A pump in a water treatment plant has a probability of $0.035$ of failing during a continuous $24\text{-hour}$ operation. What is the probability that the pump will operate successfully (not fail) during this period?

An engineer is choosing a steel section for a beam. Based on preliminary analysis, the probability of selecting an I-beam is $0.65$, and the probability of selecting a hollow structural section (HSS) is $0.25$. Since a beam cannot be both an I-beam and an HSS simultaneously, what is the probability that either an I-beam or an HSS is selected?

A batch of precast concrete panels is inspected. The probability that a panel has a minor surface defect (Event $S$) is $0.12$. The probability that it has dimensional inaccuracies (Event $D$) is $0.08$. The probability that it has both defects simultaneously is $0.03$. What is the probability that a randomly chosen panel has at least one of these defects?

A critical alarm system in a facility relies on two independent sensors, Sensor A and Sensor B. The probability that Sensor A functions correctly during an emergency is $0.98$, and the probability that Sensor B functions correctly is $0.95$. What is the probability that both sensors function correctly during an emergency?