Question

2. According to a 2018 survey conducted by the National Center for Health Statistics, 11.2 percent of the U.S. residents (ages 12 and older) have used illicit drugs in 2017. Half of the U.S. residents (ages 12 or older) in 2017 were women and 8.8 percent of women used illicit drugs in 2017. About 61 percent of the U.S. residents (ages 12 or older) in 2017 were non-Hispanic whites and 11.6 percent of the non-Hispanic whites have used illicit drugs in 2017. Please answer the following questions based on the information given in this problem.

1. If a random (12 year or older) U.S. resident happens to be non-user of illicit drugs, what is the probability that he is a man? Please show the necessary steps.   [3 points]        
2. What is the probability that a random (12 year or older) U.S resident will be a man or illicit drug user? Please show the necessary work.          [2 points]
3. What is the probability that a random U.S resident (12 year or older) will be neither a woman and nor illicit drug user? Show the necessary work.                                [2 points]
4. Are being a man and using illicit drugs independent events for a (12 year or older) U.S resident? Please show how you arrived at your answer.                                [2 points]
5. What is the probability that a random (12 year or older) U.S resident will be both non-white and illicit drug user? Assume that all the other races except the non-Hispanic whites are counted as non-white. Please show the necessary steps. [3 points]

Answer 1

Given,

P(User) = 0.112 P(Non-User) = 1 - 0.112 = 0.888

P(Women) = 0.5 P(Men) = 1 - 0.5 = 0.5

P(User | Women) = 0.088 P(NonUser | Women) = 1 - 0.088 = 0.912

P(non-Hispanic whites) = 0.61

P(User | non-Hispanic whites) = 0.116

a.

Using Bayes theorem,

P(Women | Non-user) =  P(NonUser | Women) P(Women) / P(Non-User)

= 0.912 * 0.5 / 0.888

= 0.5135135

P(Men | Non-user) = 1 - P(Women | Non-user) = 1 - 0.5135135 = 0.4864865

b.

Using Bayes theorem,

P(Non-user | Men) = P(Men | Non-user) P(Non-User) / P(Men) = 0.4864865 * 0.888 / 0.5 = 0.864

P(User | Men) = 1 - P(Non-user | Men) = 1 - 0.864 = 0.136

P(Men or User) = P(Men) + P(User) - P(Men and User)

= P(Men) + P(User) - P(User | Men) P(Men)

= 0.5 + 0.112 - 0.136 * 0.5

= 0.544

c.

P(Not Women and Non-User) = P(Men and Non-User) = P(Non-user | Men) P(Men) = 0.864 * 0.5 = 0.432

d.

P(Men and User) = P(User | Men) P(Men) = 0.136 * 0.5 = 0.068

P(Men) * P(User) = 0.5 * 0.112 = 0.056

Since  P(Men and User) P(Men) * P(User) , being a man and using illicit drugs are not independent events.

e.

P(non-Hispanic whites and user) = P(User | non-Hispanic whites) P(non-Hispanic whites)

= 0.116 * 0.61

= 0.07076