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According to the most recent adult demographic census of Ohio, 22% of residents are 18 to 29 years old, 45% are between 30 and 49 years old, and 33% are 50 and older. (Age brackets A1, A2, and A3 respectively.) Of Ag4596. those who are in the Al age bracket, 5% use E-Harmony: of those in A2, 19% use E-Harmony; and of those in , Compute the joint probabilities where E an individual uses E-Harmony (It will be helpful to use a probability tree)
P(A1nE)
P(A2nE)
a) What proportion of residents use E-Harmony?
b) You receive a message from an individual using E-Harmony expressing interest. What is the probability the individual is 50 years or older?

Respuesta :

Answer:

a) [tex] P(E) = P(E|A_1) P(A_1) +P(E|A_2) P(A_2)+ P(E|A_3) P(A_3)[/tex]

Since we have all the possible values we can replace:

[tex] P(E) = 0.05*0.22 +0.19*0.45+ 0.45*0.33=0.245[/tex]

And that would be the proportion of residents who use E-Harmony

We can find the P(A1nE)  and P(A2nE) with the following formulas:

[tex] P(A_1 n E)= P(E|A_1) P(A_1) = 0.05*0.22=0.011[/tex]

[tex] P(A_2 n E)= P(E|A_2) P(A_2) = 0.19*0.45=0.0855[/tex]

b)  For this case we want this probability [tex] P(A_3 |E)[/tex]

And from the Bayes conditional probability we have this:

[tex] P(A_3 |E) =\frac{P(A_3 n E)}{P(E)}=\frac{0.45*0.33}{0.245}=0.606[/tex]

Step-by-step explanation:

Notation

[tex]A_1[/tex] represent the event resident between 18 and 29 years old

[tex]A_2[/tex] represent the event resident between 30 and 49 years old

[tex]A_3[/tex] represent the event resident is >50 yeard old

[tex] E[/tex] represent the event the residen tuse E-Harmony

From the problem we have the following probabilities:

[tex]P(A_1) = 0.22 , P(A_2) = 0.45, P(A_3) = 0.33 [/tex]

And we have conditional probabilites also given:

[tex] P(E|A_1) = 0.05,  P(E|A_2) = 0.19[/tex]

The other probability assumed since the problem is incomplete is:

[tex]P(E|A_3) = 0.45[/tex]

Part a

For this case we can use the total probability rule given by:

[tex] P(E) = P(E|A_1) P(A_1) +P(E|A_2) P(A_2)+ P(E|A_3) P(A_3)[/tex]

Since we have all the possible values we can replace:

[tex] P(E) = 0.05*0.22 +0.19*0.45+ 0.45*0.33=0.245[/tex]

And that would be the proportion of residents who use E-Harmony

We can find the P(A1nE)  and P(A2nE) with the following formulas:

[tex] P(A_1 n E)= P(E|A_1) P(A_1) = 0.05*0.22=0.011[/tex]

[tex] P(A_2 n E)= P(E|A_2) P(A_2) = 0.19*0.45=0.0855[/tex]

Part b

For this case we want this probability [tex] P(A_3 |E)[/tex]

And from the Bayes conditional probability we have this:

[tex] P(A_3 |E) =\frac{P(A_3 n E)}{P(E)}=\frac{0.45*0.33}{0.245}=0.606[/tex]

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