Which statement best describes complementary probabilities?

Which statement best describes complementary probabilities?

• A. The probability of exactly one success in a single trial
• B. The probability of at least one success by considering the probability of no success
• C. The probability that all events occur
• D. The probability of a random variable’s expected value

Answer: (B)

Complementary probabilities rely on the idea that an event and its opposite (its complement) cover all possibilities and their probabilities add up to 1. To find the chance of something happening, it’s often easier to compute the chance that it does not happen and subtract from 1. So for at least one success, you look at the chance of zero successes and do 1 minus that. For example, if you perform two independent trials with a probability p of success in each trial, the probability of no successes is (1 − p)², and the probability of at least one success is 1 − (1 − p)². This is the complementary approach. The other options describe different ideas: exactly one success isn’t about using the complement, it’s a specific counted outcome; all events occurring is about the joint probability of a conjunction; the expected value is an average outcome, not a probability of an event.

Complementary probabilities rely on the idea that an event and its opposite (its complement) cover all possibilities and their probabilities add up to 1. To find the chance of something happening, it’s often easier to compute the chance that it does not happen and subtract from 1.

So for at least one success, you look at the chance of zero successes and do 1 minus that. For example, if you perform two independent trials with a probability p of success in each trial, the probability of no successes is (1 − p)², and the probability of at least one success is 1 − (1 − p)². This is the complementary approach.

The other options describe different ideas: exactly one success isn’t about using the complement, it’s a specific counted outcome; all events occurring is about the joint probability of a conjunction; the expected value is an average outcome, not a probability of an event.