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Math Help - probability

  1. #1
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    probability

    Four bottles are filled with grade A sauce and three with grade B sauce . If three bottles are randomly selected from the seven bottles , what is the probability that both grades of sauces are in teh selection ?

    I know this can be done by drawing the tree diagram , then sum all the probabilities . I just want to know whether this can be done by combination .

    something / 7C3

    i just cant figure out that something .
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  2. #2
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    Hello, thereddevils!

    Four bottles are filled with grade A sauce and three with grade B sauce.
    If 3 bottles are randomly selected from the seven bottles,
    what is the probability that both grades of sauces are in the selection?
    There are: . {7\choose3} = 35 possible samples.


    There are two possible scenarios:

    . . (1) Two A's and one B

    . . . . P(2A,1B) \:=\:\frac{{4\choose2}{3\choose1}}{35} \:=\:\frac{18}{35}


    . . (2) One A and two B's

    . . . . P(1A,2B) \;=\;\frac{{4\choose1}{3\choose2}}{35} \:=\:\frac{12}{35}


    Therefore: . P(\text{one of each}) \;=\;\frac{18}{35} + \frac{12}{35} \;=\;\frac{30}{35} \;=\;\frac{6}{7}



    Someone else can show you the "complement" approach.
    .
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