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Math Help - Probability applications of counting principles (permutations, combinations, etc.)

  1. #1
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    Probability applications of counting principles (permutations, combinations, etc.)

    Hi everyone, I'd like some help please

    1) A basket contains 7 red apples and 4 yellow apples. A sample of 3 apples is drawn. Find the probabilities that the sample contains more red than yellow apples.

    So, I figure I have to find the sum of the probabilities that the sample has:
    -4 red apples and 3 yellow apples, 5 red apples and 2 yellow apples, 6 red apples and 1 yellow apple, and 7 and 0 right....right?

    I want to use combinations, right? Because I know that when it asked for all red apples I did (4 3) divided by (11 3)

    *pretend that those numbers are on top of each other like in combination form*



    and my second question...
    2) given a certain number of balls, of which some are blue, pick 5 at random. thie probability that all 5 are blue is 1/2. determine the original number of balls and decide how many were blue

    This one, I have no clue.

    Thanks!
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  2. #2
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    For #1
    \dfrac{\dbinom{7}{3}+\dbinom{7}{2}\dbinom{4}{1}}{\  dbinom{11}{3}~~}
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  3. #3
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    Hi Plato,

    I've been looking and trying to understand what you've written for the last hour...and i'm still not understanding. are you sure that that answer is correct? that would give me 119/165, or 72%. i thought i would add the (7 2) and (4 1), not multiply them!

    Thanks
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  4. #4
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    In order for a sample of three to contain more reds than yellows, then it contains all reds \dbinom{7}{3}
    or it contains two reds and one yellow  \dbinom{7}{2} \dbinom{4}{1} .
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  5. #5
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    Thank you so much!!

    While I'm at it do you or anyone know how to solve the second problem? I've seen tutors, etc and nobody knows! Thanks!!
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  6. #6
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    Quote Originally Posted by gobbajeezalus View Post
    [snip]
    2) given a certain number of balls, of which some are blue, pick 5 at random. thie probability that all 5 are blue is 1/2. determine the original number of balls and decide how many were blue

    This one, I have no clue.

    Thanks!
    Let number of blue ball be x and total number of balls be n. Then \displaystyle \frac{(n - 5)! x!}{n! (x - 5)!} = \frac{1}{2}.

    Your job is to solve this equation under the conditions (i) x < n and (ii) x and n are positive whole numbers. I found a solution using trial and error in about 27 seconds (using a calculator).
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