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Math Help - Probability Quests

  1. #1
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    Probability Quests

    1. A wine collection consists of  4 bottles of cabernet and  7 bottles of zinfandel. Randomly selecting  6 bottles, find the following:

    (a)  P \{\text{all cabernets are selected} \} = \frac{\binom{6}{4}}{\binom{11}{6}}

    (b)  P \{\text{at least 1 is a cabernet} \} = \frac{\binom{6}{1} + \binom{6}{2} + \binom{6}{3} + \binom{6}{4} + \binom{6}{5} + \binom{6}{6}}{\binom{11}{6}}

    Are these correct?
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  2. #2
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    Try on these.
    a) \frac {{7 \choose 2}} {{11 \choose 6}}

    b) 1 - \frac {{7 \choose 6}} {{11 \choose 6}}

    Can you explain these to us?
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  3. #3
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    Quote Originally Posted by heathrowjohnny View Post
    1. A wine collection consists of  4 bottles of cabernet and  7 bottles of zinfandel. Randomly selecting  6 bottles, find the following:

    (a)  P \{\text{all cabernets are selected} \} = \frac{\binom{6}{4}}{\binom{11}{6}}

    (b)  P \{\text{at least 1 is a cabernet} \} = \frac{\binom{6}{1} + \binom{6}{2} + \binom{6}{3} + \binom{6}{4} + \binom{6}{5} + \binom{6}{6}}{\binom{11}{6}}

    Are these correct?
    No.


    (a)  P \{\text{all cabernets are selected} \} = \frac{\binom{4}{4} \binom{7}{2}}{\binom{11}{6}} = \frac{\binom{7}{2}}{\binom{11}{6}}.


    (b) The hard way:
     P \{\text{1 is a cabernet} \} = \frac{\binom{4}{1} \binom{7}{5}}{\binom{11}{6}} .


     P \{\text{2 are cabernet} \} = \frac{\binom{4}{2} \binom{7}{4}}{\binom{11}{6}} .


     P \{\text{3 are cabernet} \} = \frac{\binom{4}{3} \binom{7}{3}}{\binom{11}{6}} .


     P \{\text{4 are cabernet} \} = \frac{\binom{4}{4} \binom{7}{2}}{\binom{11}{6}} = \frac{\binom{7}{2}}{\binom{11}{6}} .


    Now add them together.


    (b) The easy way:

    P{At least 1 is a cabernet} = 1 - P{0 are cabernet}.

     P \{\text{0 are cabernet} \} = \frac{\binom{4}{0} \binom{7}{6}}{\binom{11}{6}} = \frac{\binom{7}{6}}{\binom{11}{6}} .
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  4. #4
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    (a) If you have 6 bottles and you choose 4 cabarets, you are left with 2 extra bottles. So this is the same as  \binom{7}{2} . How many ways you can choose 2 bottles from the remaining 7 zinfandel.


    (b) Took the complement.
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