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25 March, 00:24

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.

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  1. 25 March, 03:09
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    (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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