Results 1 to 6 of 6

Math Help - Prob of a letter bag from a set of letters

  1. #1
    Newbie
    Joined
    Jun 2013
    From
    USA
    Posts
    4

    Prob of a letter bag from a set of letters

    Say I have a set of letters totaling n (say 80 for ex)
    (some letters may occur more than once in this set)

    I randomly choose r letters from n (say r= 4)
    What is the probability of seeing the letter bag
    'aabcd' (it's a bag so order doesn't matter, 'adcab' is also ok)
    Note that there are 2 a's in this bag.

    The general form of the bag is:
    k_a of letter a
    k_b of letter b
    ..
    all total to k

    Assuming that there are n_a of letters a
    n_b of letters b
    n_c of letters c
    and so on ..
    from the set of n letters

    My naive answer is
    [n_aCk_a * n_bCk_b * ... ] / nCr

    And I know this is not correct
    Can anybody help me get the correct answer for this?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    4,164
    Thanks
    762

    Re: Prob of a letter bag from a set of letters

    Hey lettertha.

    After pick something out, do you put it back (sampling with replacement) or do you keep it (sampling without replacement)?

    If you replace it then you have a multinomial distribution. If you don't replace it, you have a multi-variable hyper-geometric distribution.

    You can look at the general proofs of these distributions to understand how to derive the combinatoric identities.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,963
    Thanks
    1784
    Awards
    1

    Re: Prob of a letter bag from a set of letters

    Quote Originally Posted by lettertha View Post
    Say I have a set of letters totaling n (say 80 for ex)
    (some letters may occur more than once in this set)
    I randomly choose r letters from n (say r= 4)
    What is the probability of seeing the letter bag
    'aabcd' (it's a bag so order doesn't matter, 'adcab' is also ok)
    Note that there are 2 a's in this bag.
    The general form of the bag is:
    k_a of letter a
    k_b of letter b
    Totally disregarding reply #2, your question is totally meaningless.

    You must tell us about the distribution of the letters in the bag.

    For example, in your example the bag could contain each letter of the English alphabetic and 54 "X's".

    You must supply more detail. As written your post is meaningless.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jun 2013
    From
    USA
    Posts
    4

    Re: Prob of a letter bag from a set of letters

    Ok, here is the details:

    I have in my collection (big letter bag):
    n_a number of a's
    n_b number of b's
    n_c number of c's

    all of them sum up to 'n'
    There is NO replacement.

    I will draw 'r' rounds, one letter in each round.

    What is the prob of seeing a letter bag (small letter bag)
    which has:
    k_a of the letters a's
    k_b of the letters b's
    ...
    (assuming that k_a < n_a; k_b < n_b; etc...)
    assuming that all these k_* sum up to k such that k < r

    I am hoping to get a combinatoric formula for the prob of seeing the small letter bag

    Ex:
    Say I have in the big bag: 5 a's; 6 b's; 8c's
    I want to find the prob of seeing 2a, 2b, 3c after randomly drawing 7 letters from the big bag
    Last edited by lettertha; June 2nd 2013 at 10:46 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    4,164
    Thanks
    762

    Re: Prob of a letter bag from a set of letters

    This is known as the multinomial distribution:

    Multinomial distribution - Wikipedia, the free encyclopedia

    Take a look at the derivation of the proofs to understand the combinatoric results involved.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Jun 2013
    From
    USA
    Posts
    4

    Re: Prob of a letter bag from a set of letters

    Hmm,
    This is interesting

    Thanks a lot to 'chiro'
    It looks like the problem is definitely of 'Multivariate hypergeometric dis' since there is no replacement
    Hypergeometric distribution - Wikipedia, the free encyclopedia

    So my initial naive answer is actually correct as it matches the Probability Mass Function formula
    I guess I underestimate my combinatorial skill too much
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: April 12th 2011, 11:13 AM
  2. Replies: 3
    Last Post: September 25th 2010, 01:33 AM
  3. Decoding letters
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 28th 2009, 02:15 AM
  4. Which letter comes next?
    Posted in the Math Puzzles Forum
    Replies: 3
    Last Post: June 4th 2009, 11:54 AM
  5. String of Letters
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: September 11th 2006, 10:07 PM

Search Tags


/mathhelpforum @mathhelpforum