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.