Results 1 to 4 of 4

Math Help - Probability that a random k-digit number contains at least one 0, 1, and 2

  1. #1
    Member
    Joined
    Jan 2010
    Posts
    232

    Probability that a random k-digit number contains at least one 0, 1, and 2

    Let k\geq 3 be any given integer. What is the probability that a random k-digit number will have at least one 0, at least one 1, and at least one 2?

    I'm thinking that this could be answered via the principle of inclusion-exclusion, but I could use some clarification (or possible correction). Here's what I've got so far.

    Assuming that leading zeroes are not allowed,
    # of k-digits containing no 0ís: 9^k
    # of k-digits containing no 1ís: 8◊9^{k-1}
    # of k-digits containing no 2ís: 8◊9^{k-1}
    # of k-digits containing no 0ís and no 1ís: 8^k
    # of k-digits containing no 0ís and no 2ís: 8^k
    # of k-digits containing no 1ís and no 2ís: 7◊8^{k-1}
    # of k-digits containing no 0ís, 1ís or 2ís: 7^k

    I then applied the following:
    |A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|
    This would obtain the complement set of what I'm looking for, so this would be subtracted from 9◊10^{k-1}.

    Am I on the right track?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Traveller's Avatar
    Joined
    Sep 2010
    Posts
    162
    You are.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jan 2010
    Posts
    232
    Thanks for confirming that.

    I currently have the answer written out as follows:
    P=(9◊10^{k-1} )-[9^k+2(8◊9^{k-1})-2(8^k)-(7◊8^{k-1})+7^k]
    This probably isn't the cleanest way to show the answer. Is there a more condensed version I could use?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member Traveller's Avatar
    Joined
    Sep 2010
    Posts
    162
    There is probably no shorter formula for this case.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Digit sum & digit product of number x
    Posted in the Algebra Forum
    Replies: 1
    Last Post: January 19th 2011, 09:07 AM
  2. Replies: 2
    Last Post: January 11th 2011, 06:34 AM
  3. 4 digit PIN number - please help
    Posted in the Statistics Forum
    Replies: 4
    Last Post: February 12th 2010, 06:47 AM
  4. 10 digit number
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: November 2nd 2009, 02:51 PM
  5. 6 digit number
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 10th 2009, 09:10 AM

Search Tags


/mathhelpforum @mathhelpforum