Carlos has chosen 12 different CDs he would like to buy: 4 are rap music, 5 are country, and 3 are heavy metal. He has only enough money to

Question

Carlos has chosen 12 different CDs he would like to buy: 4 are rap music, 5 are country, and 3 are heavy metal. He has only enough money to buy 5 of them (each CD costs the same price). So he selects 5 of them at random. What is the probability that his purchase includes at least one CD from each of the three genres.

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Calantha 5 years 2021-07-22T10:15:29+00:00 1 Answers 85 views 0

Answers ( )

  1. thienhuong
    0
    2021-07-22T10:17:18+00:00
    Reply

    Answer:

    The probability is 0.7449

    Step-by-step explanation:

    Given

    n = 12 —- total

    r = 5 — selection

    Genre =\{Rap(4), Country (5), Heavy\ metal (3)\}

    Required

    Probability of buying at least 1 of each genre

    First, we calculate the total possible selection.

    To select 5 CDs from a total of 12, we use:

    ^nC_r = \frac{n!}{(n - r)!r!}

    ^{12}C_5 = \frac{12!}{(12 - 5)!5!}

    ^{12}C_5 = \frac{12!}{7!5!}

    Expand

    ^{12}C_5 = \frac{12*11*10*9*8*7!}{7!*5*4*3*2*1}

    ^{12}C_5 = \frac{12*11*10*9*8}{5*4*3*2*1}

    ^{12}C_5 = \frac{95040}{120}

    ^{12}C_5 = 792

    So, the total selection is:

    Total = 792

    To select at least 1 from each genre, there are 6 possible scenarios.

    And they are:

    \begin{array}{ccc}{Heavy\ Metal (3)} & {Rap (4)} & {Country(5)} & {3} & {1} & {1} & 2 & {1} & {2} & {2} & {2} & {1}& {1} & {2} & {2}& {1} & {3} & {1}& {1} & {1} & {3} \ \end{array}

    The possible selections for the given scenario is:

    Possible = ^3C_3* ^4C_1 * ^5C_1 +^3C_2* ^4C_1 * ^5C_2 +^3C_2* ^4C_2 * ^5C_1 +^3C_1* ^4C_2 * ^5C_2 +^3C_1* ^4C_3 * ^5C_1 +^3C_1* ^4C_1 * ^5C_3

    Using a calculator, we have:

    Possible = 1*4*5 +3*4*10 +3*6*5 +3*6*10+3*4*5+3*4*10

    Possible= 20 +120 +90 +180+60+120

    Possible = 590

    The probability is then calculated using:

    Pr = \frac{Possible}{Total}

    Pr = \frac{590}{792}

    Pr = 0.7449

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