According to the Department of Health and Human Services, 30% of 18- to 25-year-olds have some form of mental illness.

(a) What is the probability two randomly selected 18- to 25-year-olds have some form of mental illness?
(b) What is the probability six randomly selected 18- to 25-year-olds have some form of mental illness?
(c) What is the probability at least one of six randomly selected 18- to 25-year-olds has some form of mental illness?
(d) Would it be unusual that among four randomly selected 18-to 25-year-olds, none has some form of mental illness?

If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events. I will do one problem for you, to illustrate the process.

(b) .70^6

(c) This means that 1, 2, 3, 4, 5 or 6 have mental illness.

Either-or probabilities are found by adding the individual probabilities.

To calculate the probabilities in this scenario, we need to use the information provided by the Department of Health and Human Services.

(a) To find the probability that two randomly selected 18- to 25-year-olds have some form of mental illness, you can use the probability formula for independent events. If the probability of having a mental illness is 30%, then the probability of two randomly selected individuals having a mental illness would be:

P(both have mental illness) = P(first has mental illness) * P(second has mental illness)

P(both have mental illness) = 0.30 * 0.30 = 0.09

So, the probability of two randomly selected 18- to 25-year-olds having some form of mental illness is 0.09 (or 9%).

(b) Following the same logic, to calculate the probability that six randomly selected 18- to 25-year-olds have some form of mental illness, you would use the probability formula for independent events again. The probability of each individual having a mental illness remains 0.30, so the calculation would be:

P(all six have mental illness) = P(first has mental illness) * P(second has mental illness) * P(third has mental illness) * P(fourth has mental illness) * P(fifth has mental illness) * P(sixth has mental illness)

P(all six have mental illness) = 0.30^6 = 0.000729

Therefore, the probability of six randomly selected 18- to 25-year-olds all having some form of mental illness is 0.000729 (or 0.0729%).

(c) To find the probability that at least one of six randomly selected 18- to 25-year-olds has some form of mental illness, it is easier to calculate the probability of the complement event (i.e., the probability that none of the six have a mental illness) and then subtract it from 1.

P(at least one has mental illness) = 1 - P(none have mental illness)

If the probability of not having a mental illness is 1 - 0.30 = 0.70, then the calculation would be:

P(at least one has mental illness) = 1 - P(none have mental illness) = 1 - (0.70^6) = 1 - 0.1176 = 0.8824

Therefore, the probability of at least one of six randomly selected 18- to 25-year-olds having some form of mental illness is 0.8824 (or 88.24%).

(d) To determine whether it would be unusual for none of the four randomly selected 18- to 25-year-olds to have any form of mental illness, we need to compare it to the probability. Using the probability formula for independent events, the calculation would be:

P(none have mental illness) = P(first does not have mental illness) * P(second does not have mental illness) * P(third does not have mental illness) * P(fourth does not have mental illness)

P(none have mental illness) = (1 - 0.30)^4 = 0.70^4 = 0.2401

Therefore, the probability of none out of four randomly selected 18- to 25-year-olds having any form of mental illness is 0.2401 (or 24.01%). If the probability of none having a mental illness is lower than 5%, which is considered a common threshold for unusual events, then it would be considered unusual in this case. As 24.01% is greater than 5%, it would not be considered unusual for none of the four individuals to have any form of mental illness.