According to Nielsen Media Research, approximately 52% of all U.S. households have digital cable television. In addition, 84% of all U.S. households have multiple TV sets. Suppose 45% of all U.S. households have digital cable television and have multiple TV sets. A U.S. household is randomly selected.

a. What is the probability that the household does not have digital cable television or does have multiple TV sets?

b. What is the probability that the household does have digital cable television or does not have multiple TV sets?

To find the solutions to these probability questions, we can use the concepts of union and complement.

a. The probability of a household not having digital cable television or having multiple TV sets can be calculated by adding the probabilities of these two events separately and then subtracting the probability of both events happening together.

Let's denote the probability of a household having digital cable television as P(D) and the probability of a household having multiple TV sets as P(M).

According to the information given, P(D) = 0.52 (52%) and P(M) = 0.84 (84%).

The probability of a household not having digital cable television can be found by subtracting P(D) from 1 (since there are only two possibilities - having digital cable television or not having it):
P(not D) = 1 - P(D) = 1 - 0.52 = 0.48

Similarly, the probability of a household not having multiple TV sets is:
P(not M) = 1 - P(M) = 1 - 0.84 = 0.16

Now, to find the probability that the household does not have digital cable television or does have multiple TV sets, we need to add the probabilities of these two events and then subtract the probability of both events happening together:
P(not D or M) = P(not D) + P(not M) - P(not D and not M)

P(not D and not M) can be calculated by subtracting the probability of having digital cable television and not having multiple TV sets (which is given as 45%) from 1:
P(not D and not M) = 1 - P(D and not M) = 1 - 0.45 = 0.55

Now we can substitute the values into the equation:
P(not D or M) = P(not D) + P(not M) - P(not D and not M) = 0.48 + 0.16 - 0.55 = 0.09

Therefore, the probability that the household does not have digital cable television or does have multiple TV sets is 0.09.

b. Similarly, to find the probability that the household does have digital cable television or does not have multiple TV sets, we can use the same set of probabilities.

We already calculated the probability of not having digital cable television as P(not D) = 0.48 and the probability of not having multiple TV sets as P(not M) = 0.16.

To find the probability that the household does have digital cable television or does not have multiple TV sets, we can use the formula:
P(D or not M) = P(D) + P(not M) - P(D and not M)

P(D and not M) can be calculated by subtracting the probability of not having digital cable television and not having multiple TV sets (which is given as 45%) from 1:
P(D and not M) = 1 - P(not D and not M) = 1 - 0.55 = 0.45

Now we can substitute the values into the equation:
P(D or not M) = P(D) + P(not M) - P(D and not M) = 0.52 + 0.16 - 0.45 = 0.23

Therefore, the probability that the household does have digital cable television or does not have multiple TV sets is 0.23.