Probability of At Least 2 Identical Letters in 4-Letter String

In summary: What I was trying to say is that if you had a string of four letters and you randomly chose two of those letters, the probability of those two letters being different is 1/26.In summary, the probability of having at least two identical letters in a randomly selected string of four letters is about 21%.
  • #1
Seda
71
0

Homework Statement



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

Homework Equations



None, except maybe a basic idea of permutations/combinations

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 sameBut 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:
Physics news on Phys.org
  • #2
whad does a string of four letters consist? What is that?
 
  • #3
4 letters chosen at random.


aghi

iopl

hujg

futy



etc...
 
  • #4
Seda said:
Well, I can see that obviously, the total number of possibilities is 26!

No. The total number of possibilities = (number of ways to choose 1st letter) * (number of ways to choose second letter) * etc.

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

It might be easier to use 1 - (probability of no letters the same)
 
  • #5
Sorry, I meant to say 26^4, i have no idea why I said factorial...ill edit that...
 
  • #6
Seda said:
4 letters chosen at random.


aghi

iopl

hujg

futy



etc...

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.
 
  • #7
Hmm


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...
 
  • #8
Im stumped
 
  • #9
kamerling said:
It might be easier to use 1 - (probability of no letters the same)

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.
 
  • #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)
 
  • #11
True Tedjn, it's easier to conceptualize the way you put it.
 

1. What is the probability of at least 2 identical letters in a 4-letter string?

The probability of at least 2 identical letters in a 4-letter string is 0.399 or 39.9%. This means that out of all possible 4-letter combinations, approximately 40% will have at least 2 identical letters.

2. How is the probability calculated for this scenario?

The probability is calculated by first determining the total number of possible 4-letter combinations, which is 26^4 or 456,976. Then, we calculate the number of combinations that have at least 2 identical letters. This can be done by considering the different cases - 2 identical letters, 3 identical letters, and 4 identical letters - and adding up their respective probabilities. The final step is to divide the number of combinations with at least 2 identical letters by the total number of combinations to get the probability.

3. Does the order of the letters matter in this scenario?

No, the order of the letters does not matter in this scenario. As long as there are at least 2 identical letters in the 4-letter string, it will be counted as a success. For example, the strings "abbb" and "bbba" will both be considered as having at least 2 identical letters.

4. Can this probability be applied to longer strings?

Yes, this probability can be applied to longer strings. However, as the length of the string increases, the probability of at least 2 identical letters also increases. This is because the number of possible combinations increases exponentially as the string length increases.

5. How can this probability be used in real-life situations?

This probability can be used in a variety of real-life situations, such as in cryptography, genetics, and data analysis. For example, in cryptography, this probability can be used to determine the strength of a password by calculating the likelihood of guessing the correct combination of letters. In genetics, this probability can be used to predict the likelihood of certain genetic traits being passed down from parents to offspring. In data analysis, this probability can be used to understand patterns and trends in large datasets.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
1
Views
992
  • Precalculus Mathematics Homework Help
Replies
16
Views
520
  • Precalculus Mathematics Homework Help
Replies
3
Views
3K
  • Precalculus Mathematics Homework Help
Replies
23
Views
1K
Replies
3
Views
1K
  • Precalculus Mathematics Homework Help
Replies
1
Views
998
  • Precalculus Mathematics Homework Help
Replies
7
Views
654
  • Precalculus Mathematics Homework Help
Replies
12
Views
1K
  • Precalculus Mathematics Homework Help
Replies
11
Views
2K
  • Computing and Technology
2
Replies
52
Views
3K
Back
Top