Counting, probability and statistical summaries
First identify the sample space and whether order matters. Selecting a committee differs from assigning distinct offices to its members. Sampling without replacement changes the composition for later draws; replacement can restore the same probabilities when draws are independent. For a conditional probability, restrict the denominator to the event already known to have occurred.
Expected value is the probability-weighted sum of possible outcomes, not necessarily an outcome that can occur in a single trial. Variance measures squared deviation from the mean, and its population and sample formulas have different denominators. Adding a constant shifts the mean without changing spread; multiplying by a constant multiplies standard deviation by its absolute value.
Read the data question precisely. A median uses ordered positions, while a mean uses every value. Weighted averages must reflect group sizes. An estimated probability or average is not automatically a causal conclusion. In a mathematical model, report the result within its assumptions and use the stated units. Counting all favorable arrangements is not enough unless the total number of equally likely arrangements is also correct.