The route used by a driver in commuting to work contains two...

The route used by a driver in commuting to work contains two intersections with traffic signals. The probability that he must stop at the first signal is. $4,$ the analogous probability for the second signal is. $5,$ and the probability that he must stop at one or more of the two signals is $.6 .$ What is the probability that he must stop (a) At both signals? (b) At the first signal but not at the second one? (c) At exactly one signal?

The route used by a driver in commuting to work contains two intersections with traffic signals. The probability that he must stop at the first signal is. $4,$ the analogous probability for the second signal is. $5,$ and the probability that he must stop at one or more of the two signals is $.6 .$ What is the probability that he must stop
(a) At both signals?
(b) At the first signal but not at the second one?
(c) At exactly one signal?

Key Concepts

Probability Theory

This is the foundational subject that deals with quantifying uncertainty by assigning likelihoods to events. It provides tools to analyze random phenomena, including defining events, calculating probabilities, and understanding relationships among different outcomes.

Inclusion-Exclusion Principle

This principle is used to calculate the probability of the union of events by accounting for their overlap. It states that the probability of at least one event occurring is the sum of the individual probabilities minus the probability of the events intersecting, which is pivotal in the analysis of combined events.

Intersection of Events

The intersection of two events refers to the scenario where both events occur simultaneously. Recognizing and computing the probability of the intersection is crucial for determining related probabilities, such as when both conditions are met in a problem.

Complement Rule

This rule involves calculating the probability of the complement of an event, which consists of all outcomes not included in the event. It is a useful tool when determining probabilities of events like 'not happening' or 'exactly one of several events happening.'

Exclusive OR (XOR) in Probability

This concept pertains to finding the probability that exactly one of several events occurs. It involves subtracting the probability of the intersection of events from the probability of their union, thereby isolating cases where only one event happens.

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