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Probability Question

  1. Apr 17, 2008 #1
    1. The problem statement, all variables and given/known data

    What is the probability of having at least two identical letters in a randomly selected string of four letters.

    2. Relevant equations

    None, except maybe a basic idea of permutations/combinations

    3. The attempt at a solution

    Well, probability seems like an easy topic, but I'm having trouble on this one.

    First off, I'm assuming that the letters are picked simultaneously.

    Well, I can see that obviously, the total number of possibilities is 26^4.

    ANd I know I need to account for having 2 letters the same, 3 letters the same, and 4 letters the same.

    Well, I know how to do this problem easy if I was given a specific letter to worry about. The result would be like this:

    Probability = [25/26 X 25/ 25 X 1/26 X 1/26] + [25/26 X (1/26)^3] + [(1/26)^4]

    ---------------^2 the same -------------------------^ 3 the same ------ ^ all four the same

    But this would only be the answer if I was given a specific letter to worry about being the same. How do I solve when it can be any letter?
    Last edited: Apr 17, 2008
  2. jcsd
  3. Apr 17, 2008 #2
    whad does a string of four letters consist? What is that?
  4. Apr 17, 2008 #3
    4 letters chosen at random.





  5. Apr 17, 2008 #4
    No. The total number of possibilities = (number of ways to choose 1st letter) * (number of ways to choose second letter) * etc.

    It might be easier to use 1 - (probability of no letters the same)
  6. Apr 17, 2008 #5
    Sorry, I meant to say 26^4, i have no idea why I said factorial...ill edit that...
  7. Apr 17, 2008 #6
    Blahhhh, damn it, i confused it with the meaning of "letter" in my native language, because it means completely sth else, and it made no sens to me.
  8. Apr 17, 2008 #7

    How would I figure the probability of none of the letters being the same if the letters are chosen simultaneously?

    If the letters were "rolled" in order, I could easily do (26*25*24*23)/(26^4)....but 1 - that = about 21% and that seems pretty high...
  9. Apr 17, 2008 #8
    Im stumped
  10. Apr 17, 2008 #9


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    I'd say that is exactly the simplest way to solve this one, so the math is really in finding the probability that all letters are different.

    To get started, figure out the probability that 2 letters chosen at random are different. Then go on to 3 letters, and finally 4 letters.
  11. Apr 17, 2008 #10
    Well, the easiest way I find to think about probability problems is to go back to definitions, which is to say,

    probability of no letters the same = (# of strings with four letters different) / (# of total four letter strings)
  12. Apr 18, 2008 #11


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    True Tedjn, it's easier to conceptualize the way you put it.
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