Answer:
(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.
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.