Where is the error in my reasoning about palindromes?

In summary, the mistake in the reasoning is that not all possibilities are equally likely and the number of possibilities for 3 letter words is not just 26 x 26, but also depends on the likelihood of each possibility. This means that the probability of a 3 letter palindrome is not equal to the probability of a 2 letter palindrome.
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red65
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Hello everyone, I found this problem online about probability, for me, I think that to have a 2 letter palindrome is less likely because we need to have the same letter in the 2 places which gives us 26 possibilities (aa , bb, cc ....) however for words with 3 letters we have 26 possibilities for the first and the last letter times 26 possibilities for the letter in the middle (aaa,aba,aca....) unfortunately my answer is wrong, can anyone tell me where is the mistake in my reasoning?
thanks!
 
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  • #2
red65 said:
Hello everyone, I found this problem online about probability, for me, I think that to have a 2 letter palindrome is less likely because we need to have the same letter in the 2 places which gives us 26 possibilities (aa , bb, cc ....) however for words with 3 letters we have 26 possibilities for the first and the last letter times 26 possibilities for the letter in the middle (aaa,aba,aca....) unfortunately my answer is wrong, can anyone tell me where is the mistake in my reasoning?
thanks!
Not all possibilities are equally likely. In particular, ##aa## is 26 times more likely than ##aaa##. But ##aa## has the same likelihood as ##a*a##, where ##*## is any letter.
 
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  • #3
The middle letteer doesn't matter (3 letter word). Drop it and get the same as 2 letter word.
 
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  • #4
red65 said:
can anyone tell me where is the mistake in my reasoning?
For 2 letter words you are right that there are 26 possibilities so we have ## P(\text{palindrome}) = \frac{26}{Y} ##. What is Y? For 3 letter words you are right that the number on the top is 26 x 26, but what is the number on the bottom?
 
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Related to Where is the error in my reasoning about palindromes?

1. What is a palindrome?

A palindrome is a word, phrase, or sequence that reads the same backward as forward.

2. How can I identify if my reasoning about palindromes is incorrect?

You can identify if your reasoning about palindromes is incorrect by checking if the word, phrase, or sequence reads the same backward as forward. If it does not, then there may be an error in your reasoning.

3. What are some common errors in reasoning about palindromes?

Some common errors in reasoning about palindromes include mistaking a word for a palindrome when it is not, not considering punctuation or spacing in a phrase, and confusing a palindrome with a mirror word (a word that spells a different word when read backward).

4. How can I improve my understanding of palindromes?

You can improve your understanding of palindromes by practicing identifying them, reading about their properties and patterns, and studying examples of palindromes.

5. Can a phrase or sequence be a palindrome if it is not a word?

Yes, a phrase or sequence can be a palindrome even if it is not a word. As long as it reads the same backward as forward, it can be considered a palindrome.

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