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Probability theory is the branch of mathematics concerned with analyzing random events and quantifying the likelihood of various outcomes. It provides a framework for making predictions about systems and processes where chance plays a role. Understanding probability is essential for situating events in the context of uncertainty—like forecasting weather, assessing risk, or even determining the chances of getting stopped by traffic lights.

Consider the exercise where we calculate the probability of being stopped at traffic lights. Probability theory here allows students to ascertain the chances based on given percentages. Using decimal forms (0.4 and 0.7) simplifies multiplication, a common operation in probability calculations. It is crucial to comprehend that probabilities range from 0 (the event will not occur) to 1 (the event is certain to occur), with values in between representing the gradation of likelihood.

In probability, independent events are those whose outcomes do not affect each other. The probability of two independent events both occurring is the product of their individual probabilities.

In our traffic light problem, being stopped by the first light does not impact the probability of being stopped by the second light. They are independent. This is why, for part (a) of the exercise, the likelihood of being stopped at both lights can be found by simply multiplying the probabilities of each event occurring alone, i.e., 0.4 (first light) and 0.7 (second light). The outcome—being stopped at one light—does not modify the probability of being stopped at the other. This understanding is fundamental when solving problems involving multiple steps or stages that do not influence one another.

Complementary probability deals with the likelihood of an event not occurring, which is essential for understanding the full scope of possible outcomes. It can be found by subtracting the probability of the event from 1, because the sum of the probabilities of an event and its complement is always 1.

For instance, if the exercise asks for the probability of not being stopped at a light (part b), we compute the complement. If there is a 40% chance of being stopped at the first light, then there is a 60% chance (1 - 0.4) of not being stopped. Likewise, a 70% chance of being stopped at the second light means a 30% chance (1 - 0.7) of cruising through. To calculate the probability of passing both lights without stopping, we multiply their complementary probabilities together, showcasing their combined effect. This insight into complementary probability is critical for analyzing scenarios where the non-occurrence of an event is just as informative as its occurrence.