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Advanced (Or simple?) balls in a jar probability

  1. Jun 7, 2015 #1

    csb

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    If I have 8 jars, each jar contains 5 unique ball types. However, I know that I have 20 unique ball types out there. So, I have balls labelled from B1, B2, B3, ...B20 to put into 5 jars.

    Let's say

    Jar 1 : B1, B3, B4, B7, B12 (5 unique balls)

    jar 2 : B9, B7, B11, B15, B18 (5 unique balls)

    jar 3 : B20, B1, B5, B17, B13 (5 unique balls)

    jar 4 : B4, B5, B6, B3, B17 (5 unique balls)

    jar 5 : B3, B10, B2, B14, B16 (5 unique balls)

    and so on...

    My question :

    1. What is the probability to observe one particular ball type in a jar ?

    2. What is the probability to observe one particular ball type in at least n jars ?
    Is there a well-known probability distribution or combinatorial method that might relate to this ?

    For question 1, can I say that the probability is 1/5 ? Or is it 5C1/20C5 ? Or should I use more complicated thing like Stirling number of the second kind for this problem ? Totally not sure...

    For question 2, once I know the probability from question 1, I can use binomial distribution, right ?

    Thanks
     
  2. jcsd
  3. Jun 7, 2015 #2
    The answer is simple, but it is not 1/5. What if there were 5 different ball types? Or 6?
     
  4. Jun 21, 2015 #3
    I think the answer to number 1 is 1/20. the p of the ball being in a jar is 5/20 times p of picking the ball 1/5. assuming independence.

    for number two you need the answer to number one times the p that the ball is in another jar*you pick that jar*you pick that ball

    but you also need to accoount for the p that the ball is in the space again but in the same jar that you already picked, this can make my answer for number 1 incorrect so I would say it is not an easy problem. 5C1/20C5 is not right it is way to small.
     
  5. Jun 22, 2015 #4
    There 20 unique ball types in total. If you take one jar, how many unique ball types are there in it? And how many unique ball types are there in total? So, assuming that any ball type is equally likely to be in the jar, what is the probability that any given ball type is one of the types that is in the jar?
     
  6. Jun 22, 2015 #5
    The probability that any ball type is one of the ball types in the jar is 1 because they are all a ball type.

    Is the question what is the probability that a particular ball is in at least one jar?

    I thought it was; if you distribute 20 unique balls uniformly into 5 distinct jars, 5 balls in each jar, each ball put in a jar independently with p=1/20, what is the probability that you will choose one particular ball out of a jar?
     
  7. Jun 23, 2015 #6
    I'm assuming question 1 is equivalent to: I randomly select 5 out of the 20 available types. What is the chance that type 1 is one of the 5 selected?

    The answer to that question is 1 - (19C5/20C5), which is 1/4. That's because there are 20C5 ways to select the 5 types, and 19C5 ways to select the 5 types without choosing type 1.
     
  8. Jun 23, 2015 #7
    Na because that's with out replacement. you can repeat balls so it is 1-19^5/20^5 = .2262
     
  9. Jun 23, 2015 #8
    That is the right answer, but it is more easily arrived at by considering that there are 5 mutually exclusive 1/20 chances for a given ball type to be in a given jar: the answer is 5 x 1/20 = 1/4.
     
  10. Jun 23, 2015 #9
    19C5/20C5 is not correct because it doesn't consider cases such as

    B2 B2 B3 B3 B4
    B2 B2 B2 B2 B2
    or
    B2 B3 B4 B5 B5
     
  11. Jun 23, 2015 #10
     
  12. Jun 23, 2015 #11
  13. Jun 23, 2015 #12
    So binomial with p=.25
     
  14. Jun 23, 2015 #13
    not advanced
     
  15. Jun 23, 2015 #14

    WWGD

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    Gold Member

    It seems this gives you a total of 25>20 unique balls.
     
  16. Jun 23, 2015 #15
    But only 20 unique ball types. The uniqueness of balls in a jar is trivial: a single ball cannot be in more than one jar, nor in the same jar more than once. There are 8 jars in total so a total of 40 (unique) balls.

    Note that there are balls of ball type B1 for instance in both jar 1 and jar 3.
     
  17. Jun 23, 2015 #16

    WWGD

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    Gold Member

    Ah, O.K, maybe it may be clearer to use the notation ## B_{1, i_1}, B_{2,i_2},...., B_{20, i_{20}} ## to denote the ball types, where ##i_j## ranges over the number of balls of type ##j##.
     
  18. Jun 23, 2015 #17
    So you can't have the same ball type in the same jar

    but you can have repeat of ball types for each jar

    and there are 40 total different balls in 8 jars for some reason
     
  19. Jun 24, 2015 #18
    As the first part of the problem has been shown to be trivial, and the OP has not returned to show any work on the second part it is probably not worth discussing any further.
     
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