The probability a person has read a book in the past year is 0.81. The probability a person is considered a millennial is 0.28. The probability a person has read a book in the past year and is considered a millennial is 0.25 (a) Find P(Millennial | Read a Book).
(b) Find P(Not Millennial | Did Not Read a Book).
(c) Are being considered a millennial and having read a book in the past year independent events? Justify your answer mathematically.

(a) P(Millennial | Read a Book) = 0.3086

(b) P( Not Millennial | Did Not Read a Book) = 0.8421

(c)

P(Millennial and Read a Book) = P(Read a Book) × P(Millennial)

0.25 = 0.81 × 0.28

0.25 ≠ 0.2268

Since the relation doesn't hold true, therefore, being considered a millennial and having read a book in the past year are not independent events.

Step-by-step explanation:

The probability a person has read a book in the past year is 0.81.

P(Read a Book) = 0.81

The probability a person is considered a millennial is 0.28.

P(Millennial) = 0.28

The probability a person has read a book in the past year and is considered a millennial is 0.25.

P(Millennial and Read a Book) = 0.25

(a) Find P(Millennial | Read a Book)

Recall that Multiplicative law of probability is given by

P(A ∩ B) = P(B | A) × P(A)

P(B | A) = P(A ∩ B) / P(A)

For the given case,

P(Millennial | Read a Book) = P(Millennial and Read a Book) / P(Read a Book)

P(Millennial | Read a Book) = 0.25 / 0.81

P(Millennial | Read a Book) = 0.3086

(b) Find P(Not Millennial | Did Not Read a Book)

P( Not Millennial | Did Not Read a Book) = P(Not Millennial and Did Not Read a Book) / P(Did Not Read a Book)

Where

∵  P(A' and B') = 1 - P(A or B)

P(Not Millennial and Did Not Read a Book) = 1 - P(Millennial or Read a Book)

∵  P(A or B) = P(A) + P(B) - P(A and B)

P(Millennial or Read a Book) = P(Read a Book) + P(Millennial) - P(Millennial and Read a Book)

P(Millennial or Read a Book) = 0.81 + 0.28 - 0.25

P(Millennial or Read a Book) = 0.84

So,

P(Not Millennial and Did Not Read a Book) = 1 - 0.84

P(Not Millennial and Did Not Read a Book) = 0.16

Also,

∵  P(A') = 1 - P(A)

P(Did Not Read a Book) = 1 - P(Read a Book)

P(Did Not Read a Book) = 1 - 0.81

P(Did Not Read a Book) = 0.19

Finally,

P( Not Millennial | Did Not Read a Book) = P(Not Millennial and Did Not Read a Book) / P(Did Not Read a Book)

P( Not Millennial | Did Not Read a Book) = 0.16/0.19

P( Not Millennial | Did Not Read a Book) = 0.8421

(c) Are being considered a millennial and having read a book in the past year independent events? Justify your answer mathematically.

Mathematically, two events are considered to be independent if the following relation holds true,

P(A and B) = P(A) × P(B)

For the given case,

P(Millennial and Read a Book) = P(Read a Book) × P(Millennial)

0.25 = 0.81 × 0.28

0.25 ≠ 0.2268

Since the relation doesn't hold true, therefore, being considered a millennial and having read a book in the past year are not independent events.

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step-by-step explanation: