Decision and conditions
Choose an event model before combining probabilities. Independent trials allow products. An expectation is a long-run weighted average, not a guaranteed outcome.
Worked model
Each of two independent trials succeeds with probability 1/3. What is the probability of at least one success? A failure has probability 2/3. Independence lets us multiply the two failure probabilities. Subtract from one: 5/9. Before calculating: The complement is no successes in either trial.
Contrast: a different decision
Each of three independent trials succeeds with probability 1/3. What is the probability of exactly one success? There are three disjoint positions for the only success. Each arrangement has probability (1/3)(1−1/3)². The total is 4/9. Before calculating: Count which single position succeeds, then multiply success and failure probabilities.
Transfer with solution
A fair six-sided die awards $18 when a 6 appears and charges $3 otherwise. What is the expected net gain per roll, in dollars? The positive outcome contributes 18/6 = 3. The five losing outcomes contribute −3 × 5/6 = −5/2. The expected gain is 1/2. Before calculating: Use signed gains and weight each by its probability.
Check the boundary
Exactly one success differs from at least one success. Independent events A and B have probabilities 1/3 and 1/2. Quantity A: P(A | B). Quantity B: P(A and B). B has positive probability, so the conditional probability is defined. Independence gives P(A and B) = 1/6, while P(A | B) = 1/3. The conditional probability is twice the positive joint probability; Quantity A is greater.
Each of two independent trials succeeds with probability 1/3. What is the probability of at least one success?
- A failure has probability 2/3.
- Independence lets us multiply the two failure probabilities.
- Subtract from one: 5/9.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GRE Quantitative Reasoning overview