Page 1 of 2 12 LastLast
Results 1 to 15 of 18
  1. #1
    5 Star Lounger Ruff_Hi's Avatar
    Join Date
    Feb 2001
    Location
    New York, New York, USA
    Posts
    768
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Balls and Buckets

    There are 5 buckets labeled A, B, C, D and E and 5 balls labeled a, b, c, d and e. Given that each bucket contains 1 and only 1 ball, what is the probability that ball "a" ends up in buckets "A" if none of the other buckets contain balls with a matching letter?
    (Location Australia, then UK, but now USA. Heart, outlook, attitude, etc always Australian)
    Quote: "All Happiness is the release of internal pressure"

  2. #2
    5 Star Lounger
    Join Date
    Aug 2001
    Location
    Confoederatio Helvetica
    Posts
    602
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Balls and Buckets

    <img src=/S/confused.gif border=0 alt=confused width=15 height=20>Is it <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">relevant what happens in the other buckets</span hide> <img src=/S/question.gif border=0 alt=question width=15 height=15> The probability that bucket A contains ball a is <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold"> 1 in 5 </span hide> or <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">20</span hide>% That is depending upon how the balls ended up in the bucket - For example if I asked John Grey to put ball a in bucket A the probability would <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">increase to 100</span hide>% n'est pas?

  3. #3
    Plutonium Lounger
    Join Date
    Mar 2002
    Posts
    84,353
    Thanks
    0
    Thanked 29 Times in 29 Posts

    Re: Balls and Buckets

    <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9 / 53</span hide>

  4. #4
    WS Lounge VIP sdckapr's Avatar
    Join Date
    Jul 2002
    Location
    Pittsburgh, Pennsylvania, USA
    Posts
    11,225
    Thanks
    14
    Thanked 342 Times in 335 Posts

    Re: Balls and Buckets

    I get the same as Hans <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9/53</span hide>

    Steve

  5. #5
    Platinum Lounger
    Join Date
    Nov 2001
    Location
    Melbourne, Victoria, Australia
    Posts
    5,016
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Balls and Buckets

    <P ID="edit" class=small>(Edited by AlanMiller on 02-Dec-04 08:55. )</P>I make it <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">1/4</span hide>, but I haven't woken up yet. <img src=/S/doze.gif border=0 alt=doze width=15 height=15> <img src=/S/yawn.gif border=0 alt=yawn width=15 height=15>

    Now, having shaken some cranial cells, I think the wording needs some qualification. Is the question asking:

    o What is the probability that ball "a" ends up in buckets "A" given that none of the other buckets contain balls with a matching letter?
    or
    o What is the probability that ball "a" ends up in buckets "A" and none of the other buckets contain balls with a matching letter?

    Alan

  6. #6
    4 Star Lounger
    Join Date
    Sep 2002
    Location
    Stafford, Staffordshire, England
    Posts
    585
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Balls and Buckets

    I make it <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">0.075</span hide>
    <font color=blue><font face="Script MT Bold"><big>Rob</big></font face=script></font color=blue>

  7. #7
    WS Lounge VIP sdckapr's Avatar
    Join Date
    Jul 2002
    Location
    Pittsburgh, Pennsylvania, USA
    Posts
    11,225
    Thanks
    14
    Thanked 342 Times in 335 Posts

    Re: Balls and Buckets

    I fail to see the difference in the 2 statements?

    I determined (as well as Hans, who posted before me) that there were <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">53</span hide> possibilities where none of B-E had the matching ball. In <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9</span hide> of them A had its matching ball.

    Steve

  8. #8
    Platinum Lounger
    Join Date
    Nov 2001
    Location
    Melbourne, Victoria, Australia
    Posts
    5,016
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Balls and Buckets

    The difference being conditional probability, and the two represent different numbers of possible configurations. One asks "Given that none of the buckets B,C,D,E contain their corresponding balls, what is the probability that A contains a?" The other asks "Of all the possible configurations, what is the probability that A contains a, and none of the buckets B,C,D,E contain their corresponding balls?" They are two quite different event spaces, the first being a subset of the second. It's the "if" qualifier in the question that needs further qualification.

    Your solution is for the first situation, which is the way I'd now interpret the question (with the "if" indicating "given that").

    Alan

  9. #9
    Plutonium Lounger
    Join Date
    Mar 2002
    Posts
    84,353
    Thanks
    0
    Thanked 29 Times in 29 Posts

    Re: Balls and Buckets

    I also read the if as "given that".

  10. #10
    WS Lounge VIP sdckapr's Avatar
    Join Date
    Jul 2002
    Location
    Pittsburgh, Pennsylvania, USA
    Posts
    11,225
    Thanks
    14
    Thanked 342 Times in 335 Posts

    Re: Balls and Buckets

    If I relook at your descriptions: are the 2 possibilities:

    So your "first" statement is the <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9</span hide> possibilities of A having a and none of the others having a match out of the "SUBSET" of <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">53</span hide> where none of the others was matching.

    And your "second" statement is the <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9</span hide> possibilities of A having a and none of the others having a match out of the "TOTAL" set of <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">120</span hide> possibilities.

    Am I interpreting that correct?

    As Hans mentioned, I saw the first based on "if none" as "given".

    So much "subtlety" and "interpretation" in language...
    Steve

  11. #11
    5 Star Lounger Ruff_Hi's Avatar
    Join Date
    Feb 2001
    Location
    New York, New York, USA
    Posts
    768
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Balls and Buckets

    Sorry, I cannot help you with the clarification. I lifted this from a multiple choice exam that some students were discussing. The model solution was 20% which everyone disagreed with but they hadn't settled on an answer. I worked out the same answer as Hans / Steve after reading the 'if' as 'given that'. I don't think it a is valid interpretation reading 'if' as 'and'.
    (Location Australia, then UK, but now USA. Heart, outlook, attitude, etc always Australian)
    Quote: "All Happiness is the release of internal pressure"

  12. #12
    WS Lounge VIP sdckapr's Avatar
    Join Date
    Jul 2002
    Location
    Pittsburgh, Pennsylvania, USA
    Posts
    11,225
    Thanks
    14
    Thanked 342 Times in 335 Posts

    Re: Balls and Buckets

    The solution is "close to" 20%.

    If none of the buckets holds its "match" there are 44 possibilities. Each ball is in a bucket 11 times. If you "try to extend" the logic and say therefore the should be 11 cases with "a" being in A to make 11/(44+11) = 11/55 = 1/5 = 20%

    The reality is that there are only <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9</span hide> cases where A has a and the others have none of the match to give <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">9/(44+9) = 9/53 = 17.0%</span hide>

    Steve

  13. #13
    5 Star Lounger Ruff_Hi's Avatar
    Join Date
    Feb 2001
    Location
    New York, New York, USA
    Posts
    768
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Balls and Buckets

    Well done Steve - I think you have solved it both ways (correct answer and faulty logic to show how they got to 20%).
    (Location Australia, then UK, but now USA. Heart, outlook, attitude, etc always Australian)
    Quote: "All Happiness is the release of internal pressure"

  14. #14
    3 Star Lounger
    Join Date
    Nov 2004
    Location
    Nottingham, Nottinghamshire, United Kingdom
    Posts
    326
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Balls and Buckets

    And I agree with absolutely everything that Alan says! Oh yes. <img src=/S/confused3.gif border=0 alt=confused3 width=45 height=45>
    <font color=448800><font face="Comic Sans MS"><big>Lyra J </font color=448800></font face=comic></big>
    <img src=/S/flags/UK.gif border=0 alt=UK width=30 height=18> Ducking the arrows in Robin Hood country <IMG SRC=http://www.wopr.com/w3tuserpics/Lyra_J_sig.gif ALT="No, Admins, no! I'm sorry, okay!" title="No, Admins, no! I'm sorry, okay!">

  15. #15
    5 Star Lounger
    Join Date
    Aug 2001
    Location
    Confoederatio Helvetica
    Posts
    602
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Re: Balls and Buckets

    Hey <img src=/S/exclamation.gif border=0 alt=exclamation width=15 height=15> Didn't I get there first <img src=/S/question.gif border=0 alt=question width=15 height=15> <img src=/S/blowup.gif border=0 alt=blowup width=60 height=60>

Page 1 of 2 12 LastLast

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •