According to the most recent adult d demographic census of Ohio, 24% of resident

Question

According to the most recent adult d demographic census of Ohio, 24% of residents are 18 at 29 years old, 47% are between 30 and 49 years old, and 29% are 50 and older. (Age brackets A_1, A_2, and A_3 respectively) Of those who are in the A_1 age bracket, 7% use E-Harmony; of those in A_2, 21% use E-Harmony; and of those in A_3, 47%. Compute the joint probabilities where E = an individual uses E-Harmony (It will be helpful to use a probability tree) What proportion of residents use E-Harmony? You receive a message from an individual using E-Harmony expressing interest. What is the probability the individual is 50 years or older?

Explanation / Answer

Using the given information, P(A1 intersection E)=P(A1)*P(A1)*P(E) [use the multiplication rule, P(A intersection B)=P(A)*P(B), for independent events]

=0.24*0.07

=0.0168 [ans]

P(A1 intersection E bar)=0.24*0.93=0.2232 [P(Ebar) denotes proportion of 18 to 29 years old who do not use E-Harmony] [ans]

Similarly, P(A2 intersection E)=0.47*0.21=0.0987 [ans]

P(A2 intersection E bar)=0.47*0.79=0.3713 [ans]

P(A3 intersection E)=0.29*0.47=0.1363 [ans]

P(A3 intersection E bar)=0.29*0.53=0.1537 [ans]

a. P(residents using E-Harmony]=P(A1 intersection E)+P(A2 intersection E)+P(A3 intersection E)

=0.0168+0.0987+0.1363

=0.2518 [ans]

b. Use Bayes theorem to compute the following probability.

P(A3|E)=P(A3)*P(E|A3)/{P(A3)*P(E|A3)+P(A2)P(E|A2)+P(A1)P(E|A1)}

=0.29*0.1363/{0.29*0.1363+0.47*0.3713+0.24*0.0168}

=0.1813 [ans]

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