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    Question

    A poly-bag contains (3a - 4) red, (2a - 3) white, and (a

    - 1) black rubber bands. Two rubber bands are drawn randomly from the bag, and the probability that both are red is 7/30. Based on this information, determine the probability that at most one of the two drawn rubber bands is white.
    A 14/11 Correct Answer Incorrect Answer
    B 11/12 Correct Answer Incorrect Answer
    C 12/13 Correct Answer Incorrect Answer
    D 16/17 Correct Answer Incorrect Answer
    E None of these Correct Answer Incorrect Answer

    Solution

    ATQ,

    Number of rubber bands in the Poly-bag = 3x - 4 + 2x - 3 + x - 1 = 6x - 8

    ATQ:

    {(3a-4)/(6a-8) × (3a-5)/(6a-9)} = 7/30

    Or, 30 X (9a² - 27a + 20) = 7 X (36a² - 102a + 72)

    Or, 270a² - 810a + 600 = 252a² - 714a + 504

    Or, 18a² - 96a + 96 = 0

    Or, 3a² - 16a + 16 = 0

    Or, (3a - 4) (a - 4) = 0

    So, 'a' = (4/3) or 'a' = 4

    But 'a' should be an integer. So, 'a' = 4

    Number of white rubber bands = 2 X 4 - 3 = 5

    Total number of rubber bands = 6 X 4 - 8 = 16

    Required probability:

    Probability that none of the rubber bands is white + Probability that one of the rubber bands is white

    = (11/16) X (10/15) + 2 X (5/16) X (11/15)

    = (110 + 110) ÷ 240

    = (220/240) = (11/12)

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