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Suppose in total there are $500$ meaningful three letter words formed from the letters from $A,B,...,Z$. Consider a sequence of $100$ random letters. We are interested in the occurrence of three consecutive letters in the sequence which form a meaningful three letter word. What is the probability that the sequence contains at least one such meaningful three letter word?

I totally have no idea how to start this problem. Can anyone give some hints?

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1)Find the number of 3-letter words in a 100 letter sequence.

Consider: (1,2,3) as a set, (2,3,4) as a set etc. until (98,99,100)

Therefore there are 98 3-letter sequential words in a 100 letter sequence.

2)Possible permutations of the first 3 letter sequence

Consider: 26 choices for the first letter, 26 choices for the second letter and 26 choices for the third letter.

Therefore 26x26x26 = 17576 possibilities.

3)Possibility of a meaningful word in the first 3 letter sequence

Consider: Total of 500 meaningful 3 letter words out of 17576 possibilities

Therefore 500/17576 = 2.84% chance for the first 3 letters.

4)Possibility of a non-meaningful word

1-2.84%=97.16%

5)Possibility of zero meaningful words

(97.16%)^98 = 0.059 since there are 98 3 letter words

6)Possibility of at least one meaningful word

1-possibility of zero meaningful words = 1 - 0.059 = 0.94 = 94%

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Hint: how many sequences of three consecutive letters are there in a single 100-letter sequence?

After that: take the probability of a three-letter word being meaningful. Do you then know how to arrive at the probability of at least one of your list of three-letter words being meaningful?

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  • $\begingroup$ This doesn't quite do it -- yes, you can count the number of three-letter sequences in a 100-letter sequence, but those three-letter sequences are not independent of each other. $\endgroup$ – Nick Peterson Nov 3 '13 at 15:06
  • $\begingroup$ I didn't imply independence. Count in steps of $3$ from $0$, $1$, and $2$ (to $99$). Added `consecutive' to make my post clearer. $\endgroup$ – Newb Nov 3 '13 at 15:07
  • $\begingroup$ @Newb: I still not understand. Can illustrate using some simple examples? $\endgroup$ – Idonknow Nov 4 '13 at 2:36
  • $\begingroup$ Suppose you get the hundred-letter sequence $X_1 \ldots X_{100}$. Then how many consecutive three-letter sequences are there? There's the sequence $X_1, X_2, X_3$. There's $X_2, X_3, X_4$. There's $X_3, X_4, X_5$ and so on. $\endgroup$ – Newb Nov 4 '13 at 3:51

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