According to Nielsen Media Research, approximately 86 % of...

According to Nielsen Media Research, approximately $86 \%$ of all U.S. households have high-definition television (HDTV). In addition, $49 \%$ of all U.S. households own digital video recorders (DVR). Suppose $40 \%$ of all U.S. households have HDTV and have DVR. A U.S. household is randomly selected. a. What is the probability that the household has HDTV or has DVR? b. What is the probability that the household does not have HDTV or does have DVR? c. What is the probability that the household does have HDTV or does not have DVR? d. What is the probability that the household does not have HDTV or does not have DVR?

According to Nielsen Media Research, approximately $86 \%$ of all U.S. households have high-definition television (HDTV). In addition, $49 \%$ of all U.S. households own digital video recorders (DVR). Suppose $40 \%$ of all U.S. households have HDTV and have DVR. A U.S. household is randomly selected.
a. What is the probability that the household has HDTV or has DVR?
b. What is the probability that the household does not have HDTV or does have DVR?
c. What is the probability that the household does have HDTV or does not have DVR?
d. What is the probability that the household does not have HDTV or does not have DVR?

Key Concepts

De Morgan's Laws

These laws provide relationships between unions and intersections of events in the context of complements. They state that the complement of the union of two events is the intersection of their complements, and similarly, the complement of the intersection is the union of their complements. This is valuable for transforming and simplifying probability problems involving 'not' operations.

Complement Rule

This is a fundamental concept in probability whereby the probability of an event not occurring is equal to one minus the probability that it does occur. This rule is particularly useful in simplifying probability calculations by allowing one to work with the complement of an event rather than the event itself.

Addition Rule for Probability

This rule is used to calculate the probability that at least one of two events occurs. It states that the probability of the union of two events is equal to the sum of their individual probabilities minus the probability of their intersection, thereby correcting for any overcounting of outcomes that are common to both events.

Inclusion-Exclusion Principle

This principle provides a systematic way to calculate the probability of unions of events by adding the probabilities of individual events and subtracting the probabilities of their intersections. It ensures that any overlap between events is not counted more than once, which is especially useful when dealing with multiple events.

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