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Thread: Check Answers - Outcomes with colored balls

  1. #1
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    Exclamation Check Answers - Outcomes with colored balls

    An urn contains 5 red, 4 white, and 3 blue numbered balls. We draw 3 together at random, with no order on the drawn balls.
    i) How many possible outcomes are there?
    ii) How many outcomes have one ball of each color?
    iii) How many outcomes have at least 2 balls of the same color?
    iv) If the drawing is uniformly random, about what percent of the time will we get at least 2 white balls?

    i) 12 choose 3 = 220
    ii) (5 choose 1)*(4 choose 1)*(3 choose 1) = 5*4*3 = 60
    iii) (5 choose 2)+(5 choose 3)+(4 choose 2)+(4 choose 3)+(3 choose 2)+(3 choose 3) = 10+10+6+4+3+1 = 34 outcomes

    iv) I made a table where k = 0,1,2,3 and k=#of white balls drawn
    k 0 1 2 3
    Fraction 56/220 112/220 48/220 4/220
    % 25.4 50.9 21.8 1.8

    So I added 21.8% + 1.8% to get 23.6%
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  2. #2
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    Re: Check Answers - Outcomes with colored balls

    all the balls are numbered or just the blue ones?
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  3. #3
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    Re: Check Answers - Outcomes with colored balls

    i) and ii) look correct.
    For iii), you are occasionally only choosing 2 balls. But, if there are at least two balls of the same color, then there is not one ball of each color. You have 220 possible ways of drawing 3 balls and 60 of those ways result in a ball of each color being drawn, so there should be 220-60 = 160 possible ways to draw 2+ balls of the same color.

    Let's check to make sure that is correct. Let's find all disjoint cases:
    2 red balls, one non-red ball: $\binom{5}{2}\binom{7}{1} = 70$
    3 red balls: $\binom{5}{3} = 10$
    2 white balls, one non-white ball: $\binom{4}{2}\binom{8}{1} = 48$
    3 white balls: $\binom{4}{3} = 4$
    2 blue balls, one non-blue ball: $\binom{3}{2}\binom{9}{1} = 27$
    3 blue balls: $\binom{3}{3} = 1$
    Sum is 160, just as expected.

    You have 4 correct.
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  4. #4
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    Re: Check Answers - Outcomes with colored balls

    Sorry that question was worded poorly. All balls are numbered, as a way to distinguish one from another but they are still just that, balls.
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