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Math Help - probably wrong

  1. #1
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    Thumbs down probably wrong

    Hi folks,

    This probability question has me puzzled!

    In order to start a game of chance a player throws a dice until he obtains a 6. He then records whatever score he obtains on subsequent throws. For example: Throws of 2,4,3,6,4,6,2,5 give recorded scores of 4,6,2,5.

    Calculate the probability that
    a) the first score recorded is that of the player's fourth throw
    b) the player does not record a score in his first 5 throws

    The player has 7 throws. Calculate the probability that he will have recorded

    c) exactly three 5s and a 3

    d) a total score of 3


    So:The first bit is ok. If the first score recorded is on the 4th throw, then the 3rd throw must have been a six. For this to happen the first throw and second throw must not be six, so

    P(6 on 3rd throw) = 5/6. 5/6. 1/6 = 25/216

    The second bit is where I get stuck. The problem is with the wording. Does this mean that I am to assume a six is thrown on the 5th try, so that scores can be recorded from the 6th throw onwards or do I just assume that the first 5 throws do not produce a six?

    i.e. P(no 6 in 5 throws) = 5/6.5/6.5/6.5/6.1/6.

    or = 5/6.5/6.5/6.5/6.5/6


    in fact neither give the correct answer! Any suggestions here?


    For part c we have to calculate the probability of throwing:
    x.x.6.3.5.5.5. where x is between 1 and 5
    x.x.6.5.3.5.5.
    x.x.6.5.5.3.5
    x.x.6.5.5.5.3


    P(score of 3) = 5/6.5/6.1/6.1/6.1/6.1/6.1/6.
    since selecting any number on a dice is a 1/6. Then multiply by 4 to cover all the permutations. This gives the correct answer but now I am stuck on d!

    To score 3, I reckon we have the following possibilities:


    x.x.x.x.x.6.3 i.e. 5/6.5/6.5/6.5/6.5/6.1/6.1/6.

    x.x.x.x.6.2.1 ditto

    x.x.x.x.6.1.2 ditto

    x.x.x.6.1.1.1 ditto.



    Then add up the probabilities.This gives the wrong answer and makes me doubt whether my approach in part c is correct.



    Any help with this would be appreciated!


    best regards
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  2. #2
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    Quote Originally Posted by s_ingram View Post
    In order to start a game of chance a player throws a dice until he obtains a 6. He then records whatever score he obtains on subsequent throws. For example: Throws of 2,4,3,6,4,6,2,5 give recorded scores of 4,6,2,5.

    Calculate the probability that
    a) the first score recorded is that of the player's fourth throw
    b) the player does not record a score in his first 5 throws
    The correct amswer is \left( {\frac{5}{6}} \right)^4 .
    Notice that none of the first four is a six.
    Even if the fifth throw is a six it still is not recorded.
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  3. #3
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    Thanks Plato, that is the correct answer. I felt I had to account for the 5th dice. Now I see that I don't!

    As you saw I got the correct answer for part c but I am still stuck on part d. Any idea how to do that bit?
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  4. #4
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    Quote Originally Posted by s_ingram View Post
    I am still stuck on part d. Any idea how to do that bit?
    For the part (d), there are only three cases to consider.
    1) He throws a first six on the fourth throw; then throws three ones.
    2) He throws a first six on the fifth throw; then throws a one and a two or two and one.
    3) He throws a first six on the sixth throw; then throws a three.
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  5. #5
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    Exactly! And from my post you can see that I lined up the cases and showed how I calculated the probabilities:

    So, P(x.x.x.x.x.6.3) = 5^5/6^7 (the x means anything except 6)

    i.e. 5/6 for each x and 1/6 for a six or a 3 or a 5. Sorry about the notation here I haven't got round to LaTex yet!

    Then P(x.x.x.x.6.1.2) = P(x.x.x.x.6.2.1) = 2 x 5^4 / 6^7

    And P(x.x.x.6.1.1.1) = 3. 5^3 / 6^7.

    My answer is 5^5 / 6^7 + 2.5^4 / 6^7 + 3.5^3 / 6^7

    And like I say, I don't get the right answer (which is 5^3 / 6^5).
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  6. #6
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    Quote Originally Posted by s_ingram View Post
    And like I say, I don't get the right answer (which is 5^3 / 6^5).
    First P(xxx6111) = \frac{{5^3 }}{{6^7 }}.

    Then \frac{{5^5 }}<br />
{{6^7 }} + \frac{{2 \cdot 5^4 }}<br />
{{6^7 }} + \frac{{5^3 }}<br />
{{6^7 }} = \frac{{5^3 }}<br />
{{6^7 }}\left( {25 + 10 + 1} \right) = \frac{{5^3 }}<br />
{{6^5 }}
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  7. #7
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    Ahh!!! Now I see. Thanks, Plato.
    Sorry just noticed something! In part c
    P(x.x.6.3.5.5.5) = 4C3 5/6. 5/6. 1/6. 1/6. 1/6. 1/6. 1/6.
    The 4C3 is because there are three 5s. For this reason I did the same with the three 1s when calculating P(x.x.x.6.1.1.1) that's why I was getting the wrong answer. So, may I ask you dear Plato, why is the 4C3 included in the first case but not in the second. It's details like this that make all the difference!
    Last edited by s_ingram; June 28th 2009 at 04:56 AM. Reason: noticed something previously overlooked
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  8. #8
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    Quote Originally Posted by s_ingram View Post
    Ahh!!! In part c P(x.x.6.3.5.5.5) = 4C3 5/6. 5/6. 1/6. 1/6. 1/6. 1/6. 1/6. The 4C3 is because there are three 5s. For this reason I did the same with the three 1s when calculating P(x.x.x.6.1.1.1) that's why I was getting the wrong answer. Why is the 4C3 included in the first case but not in the second.
    The string "3555" can be arranged in \frac{4!}{3!} ways.
    There is only one way to arrange "111"
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