Solved Problems

A structural engineer needs to select 3 steel beams for testing from a batch of 10 beams. In how many ways can the beams be selected?

A construction project has 4 distinct phases (Site Prep, Foundation, Structure, Finishes). In how many different orders can these phases be scheduled if there were no dependencies (purely hypothetical)?

An engineering firm analyzes two types of defects in concrete blocks: surface cracking (Event $A$) and dimensional inaccuracy (Event $B$). Based on historical data, the probability of surface cracking is $0.05$, the probability of dimensional inaccuracy is $0.08$, and the probability of a block having both defects is $0.02$. What is the probability that a randomly selected concrete block has at least one of these defects?

A structural component is considered safe if it passes three independent stress tests. The probabilities of passing the tests are $0.98$, $0.95$, and $0.99$, respectively. What is the probability that the component fails at least one of the tests?