Results 1 to 3 of 3

Math Help - Problem from my test! Help!

  1. #1
    Newbie
    Joined
    Dec 2007
    Posts
    2

    Problem from my test! Help!

    Hey have a look at these

    You have 2 standard 52 card decks of cards that have been shuffled randomly. You turn over the first card from the first deck, and the first card from the second deck,
    (i) What is the probability they match?
    (ii)You keep turning over the top card on each deck, until you have gone through the whole deck. How many matches do you expect to find?
    (iii) You have a thousand letters and envelopes addressed to people, the envelopes get mixed up and the letters get randomly put into envelopes. How many do you expect to end up in the right envelope?

    I thought there was a 1/52 chance for the first part but the other parts were confusing. Can anyone help? I'd really appreciate it
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,914
    Thanks
    779
    Hello, konix!

    I hope this wasn't a homework problem.
    The theory required is quite advanced.
    . . In fact, I won't into the details here . . .


    You have 2 standard 52 card decks of cards that have been shuffled randomly.
    You turn over the first card from the first deck, and the first card from the second deck.
    (i) What is the probability they match?

    (ii)You keep turning over the top card on each deck, until you have gone through
    the whole deck. How many matches do you expect to find?

    (iii) You have a thousand letters and envelopes addressed to people,
    the envelopes get mixed up and the letters get randomly put into envelopes.
    How many do you expect to end up in the right envelope?
    (i) The first card can be any card: . \frac{52}{52} \:=\:1
    . . .The other card must match: . \frac{1}{52}
    The probability that the two top cards match is: . (1)\left(\frac{1}{52}\right) \:=\:\frac{1}{52} . You were right!


    {ii) and (iii) This is a classic (very old) problem.
    It has been called "The Mis-addressed Envelopes" or "The Careless Hatcheck Girl".
    (Make up your own scenarios.)

    A permutation of objects in which no object is in its proper place
    . . is called a "complete permuatation" or, more commonly, a disarrangment.

    You might try to discern a pattern, but it's very elusive . . .

    . . . \begin{array}{ccc}\text{objects} & \text{disarrangements} \\<br />
1 & 0 \\ 2 & 1 \\ 3 & 2 \\ 4 & 9 \\ 5 & 44 \\ 6 & 265 \\ \vdots & \vdots \end{array}


    With a deck of 6 cards, there are 6! = 720 possible orders for the first deck.
    There are 265 orders for the second deck to have no matches.
    Hence, there are: . 720 - 265 \:=\:455 orders with some match.

    Therefore: . P(\text{at least one match}) \:=\:\frac{455}{720} \:=\:0.63194 \:\approx\:\frac{5}{8}


    With 52 cards, we need the number of possible disarrangements, d(52)
    . . and divide by 52!, the number of possible orderings of the cards.
    This is the probability of no matches.
    . . Subtract from one (1), to get the probability of some matches.

    Skipping all the underlying theory, as the number of objects gets larger,
    . . the probability of getting no matches approaches :\frac{1}{e} \:=\:0.367879441 ...


    Maybe you can apply all this to your problem . . .


    Edit: I was reminded that the proper term is derangement
    . . . . That's true . . . *sigh*
    Last edited by Soroban; December 16th 2007 at 11:35 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2007
    Posts
    2
    thanks a million, should be able to do it now
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Problem I couldn't do on the test but should be able too
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 12th 2010, 11:08 AM
  2. t-test and correlation problem
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: November 21st 2009, 01:06 PM
  3. Help with Math Problem that was on a test...
    Posted in the Calculus Forum
    Replies: 4
    Last Post: September 13th 2009, 07:14 PM
  4. need help with a sign test problem
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: June 20th 2009, 05:15 PM
  5. Take home test problem
    Posted in the Calculus Forum
    Replies: 0
    Last Post: December 4th 2008, 09:26 AM

Search Tags


/mathhelpforum @mathhelpforum