Results 1 to 2 of 2

Math Help - Cardinality of events?

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    1

    Cardinality of events?

    I do not understand how to do these problems.

    1. 1 dice is rolled five times, find the probability that the sequence is increasing (ex. 12345, 12346, 23456, etc.)
    2. 1 dice is rolled five times, find the probability that the sequence contains exactly two numbers (11234, 22314, etc.)

    I can get the cardinality of the sample space. (6^5)


    I'm mostly having a hard time with the cardinality of the events. There are so many possibilities for each of these, it's impossible to write them out on paper. How do I get the cardinality of all these events?



    This assignment is due soon and for the most part I understand it. But I do not understand how to get these problems with the large event cardinalities.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,905
    Thanks
    765
    Hello, jimpo2!

    1. A die is rolled five times.
    Find the probability that the sequence is increasing (ex. 12345, 12346, 23456, etc.)
    This one is simple if you think about it . . .

    Of the 6^5 = 7776 possible outcomes,
    . . only five are in increasing order: . \bigg\{12345,\:12356,\:12456,\:134456,\:23456\bigg  \}

    Therefore: . P(\text{increasing order}) \;=\;\frac{5}{7776}



    2. A die is rolled five times.
    Find the probability that the sequence contains exactly one pair: (11234, 22314, etc.)
    The Pair can be any of the 6 values.

    The Pair can show up in any two of the five rolls: . {5\choose2} = {\color{blue}10} ways.

    The probability that two rolls have that Pair is: . \left(\frac{1}{6}\right)^2 \:=\:{\color{blue}\frac{1}{36}}

    The other three rolls must not match the Pair or each other: . \frac{5}{6}\cdot\frac{4}{6}\cdot\frac{3}{6} \;=\;{\color{blue}\frac{5}{18}}


    Therefore: . P(\text{one Pair}) \;=\;6\cdot 10\cdot\frac{1}{36}\cdot\frac{5}{18} \;=\;\frac{25}{54}

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cardinality
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: December 7th 2010, 11:01 AM
  2. Cardinality of a set
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: April 29th 2010, 05:03 PM
  3. Cardinality
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: January 1st 2010, 12:28 PM
  4. cardinality
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 6th 2008, 03:44 PM
  5. cardinality
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: April 15th 2008, 08:18 PM

Search Tags


/mathhelpforum @mathhelpforum