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Math Help - Combinatorial Probability

  1. #1
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    Combinatorial Probability

    There are two exercises in my probability and statistics textbook with which I am experiencing difficulty.

    Exercise 1-28 A high school lottery uses two sets of numbered balls. One set consists of ten white balls numbered 1-10 and the second set contains twenty blue balls numbered 1-20. To play, you select two white balls and two blue balls.

    (b) Your lottery ticket consists of four numbers: two white numbers, each between 1 and 10 inclusive, and two blue numbers, each between 1 and 20, inclusive. What is the probability that your lottery ticket contains exactly one matching white number and two matching blue numbers?

    I know the total number of outcomes is 10^2 \times 20^2 = 40000. How would I calculate the number of outcomes resulting in exactly one matching white number and two matching blue numbers (so I can divide that by the number of total outcomes to find the probability)?

    Exercise 1-29 At a picnic, there was a bowl of chocolate candy that had 10 pieces each of Milky Way, Almond Joy, Butterfinger, Nestle Crunch, Snickers, and Kit Kat. Jen grabbed six pieces at random from this bowl of 60 chocolate candies.

    (a) What is the probability that she got one of each variety?

    There are 60 \cdot 59 \cdot 58 \cdot 57 \cdot 56 \cdot 55 = \ _{60}P_6 total outcomes. Of those outcomes, 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = \ _6P_6 result in one of each variety. Therefore, shouldn't the solution be \frac{_6P_6}{_{60}P_6} = 1.99745 \cdot 10^{-8}? The solution in the back of book is 0.01997.
    Last edited by NOX Andrew; October 11th 2010 at 01:17 PM.
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  2. #2
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    Hello, NOX Andrew!

    You should use Combinations, not Permutations.


    1-28. A high school lottery uses two sets of numbered balls.
    One set consists of ten white balls numbered 1-10
    and the second set contains twenty blue balls numbered 1-20.
    To play, you select two white balls and two blue balls.

    What is the probability that your lottery ticket contains exactly
    one matching white number and two matching blue numbers?

    There are: . {10\choose2}{20\choose2} \:=\:45\cdot190 \:=\:8550 possible outcomes.

    We want one of the two White winners: 2 ways
    [color=beige]and both of the two Blue winners: 1 way

    Hence, there are 2 ways to have one white winner and two blue winners.

    P(\text{1 White, 2 Blue}) \;=\;\dfrac{2}{8550} \;=\;\dfrac{1}{4275}




    1-29) At a picnic, there was a bowl of chocolate candy that had:
    10 pieces each of Milky Way, Almond Joy, Butterfinger, Nestle Crunch,
    Snickers, and Kit Kat.
    Jen grabbed six pieces at random from this bowl of 60 chocolate candies.

    What is the probability that she got one of each variety?

    There are: . \displaystyle {60\choose6} \:=\:50,\!630,\!860 possible outcomes.

    To get one of each, there are: . {10\choose1}{10\choose1}{10\choose1}{10\choose1}{1  0\choose1}{10\choose1} \:=\:1,\!000,\!000 ways.

    P(\text{one of each}) \;=\;\dfrac{1,\!000,\!000}{50,\!063,\!860} \;=\;\dfrac{50,\!000}{2,503,\!193}
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