7. What is the sum of the probabilities of two complementary events? Explain why your answer makes sense.

The sum of the probabilities of two **complementary events** is **1**. ### **Explanation:** Two events are **complementary** if they are the only two possible outcomes of a particular experiment or situation, and they cannot occur simultaneously. Essentially, one event's occurrence means the other cannot occur, and vice versa. Mathematically, if \( A \) and \( A' \) are complementary events, then: \[ P(A) + P(A') = 1 \] This makes sense because the total probability of all possible outcomes in a probability space must equal 1 (representing certainty). Since complementary events cover all possible outcomes without overlap, their probabilities together account for the entire probability space. **Example:** Consider flipping a fair coin: - Let event \( A \) be getting "Heads". - The complementary event \( A' \) is getting "Tails". Since these are the only two possible outcomes: \[ P(\text{Heads}) + P(\text{Tails}) = 0.5 + 0.5 = 1 \] This demonstrates that the probabilities of complementary events always add up to 1.