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Math Help - Ambiguity in probability questions

  1. #1
    Senior Member Paze's Avatar
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    Ambiguity in probability questions

    I feel that many probability questions are very ambiguous concerning the choice of words. For example:

    In a talent show there are 12 contestants. The judges will choose 5 contestants and place them in the top 5 places. How many ways can this be done?

    I seem to stumble upon these sorts of questions all the time where I find myself asking: ''Do they want me to find out how many ways 5 contestants can be placed in 5 places?'' ''Do they want me to find out how many ways 5 contestants can emerge from 12 and then find out how many ways these 5 can be placed in the top 5 places?''.

    Maybe it's because I lack probability knowledge but is there any way to understand these explicitly?
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  2. #2
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    Re: Ambiguity in probability questions

    I think it's mostly just practicing these types of questions.

    In your example, we know there are 12 total contestants. We need to pick 1st place, 2nd place, and so on.

    How many choices of people do we have for first place? Answer: 12
    For second place we have 11 choices
    then 10 for 3rd, 9 for 4th and 8 for 5th.

    So the overall answer is: 12 * 11 * 10 * 9 *8 (or 12 \;P\; 5)

    We can also arrive at the answer a different way:

    First we choose 5 out of the 12 people \binom{12}{5} (or 12 \;C\; 5)
    then we rank them: 5 choices for 1st, 4 for 2nd, 3 for 3rd, 2 for 4th, 1 for 5th ( 5! choices total)

    So the answer we get is \binom{12}{5} * 5!

    If you check on your calculator, you will see that these two seemingly different answers are the same.

    Important questions to ask yourself:

    Does order matter?
    Are these objects distinct or indistinct?
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  3. #3
    Senior Member Paze's Avatar
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    Re: Ambiguity in probability questions

    Thanks Jame. I seem to be making the question more complicated when in fact they are just asking ''How many ways can you arrange 12 contestants in the top 5 places''.
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