If we are comparing a text to itself, we basically are mathematically finding the probability that if we choose characters from the text, the characters will be the same.
Here is the formula:
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where is the number of characters in the alphabet, is the number of times the th of the alphabet appears in the plaintext, and is the number of letters in the plaintext.
Let us try to derive this formula. Probability is defined as the number of ways you get what you want divided by the total number of possibilities. How many ways are there to choose any letters from a group of letters? It is of course, which is equal to . This is the denominator. To calculate the numerator, we first calculate the number of ways to pick a's from our plaintext and add that to the number of ways to pick b's from our plaintext, and so on. If the number of a's in our plaintext was , then the number of ways to pick a's is , this is equal to as we have shown before. This numerator and denominator gives us , the 's cancel giving us our desired formula:

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