Introduction to Probability

A random process generates outcomes that fill a sample space, and an event is a subset of it. Two dice generate 36 equally likely ordered outcomes, so the event sum at least 10 has probability $\frac{6}{36}$, while the eleven possible sums are events of very different sizes and are not equally likely. Every probability satisfies $0 \le P(A) \le 1$ and the whole sample space sums to 1, which makes both a check on any answer. The complement rule $P(A^c) = 1 - P(A)$ follows, and it is the standard route through at least one questions: with four independent parts each failing at 0.2, $P(\text{at least one}) = 1 - (0.8)^4 = 0.5904$.

Three failures recur. Probabilities are reported outside 0 to 1, or a set of outcome probabilities is left summing to something other than 1, with no check performed. Equally likely is assumed from the mere number of outcomes: two outcomes read as 50-50, three read as a third each, when the count of outcomes says only how many pieces exist, not their sizes. And at least one is computed by adding the individual probabilities, $4 \times 0.2 = 0.8$ instead of $1 - 0.8^4$, which double counts the overlaps and, with enough trials, produces an answer greater than 1.