B Where is the error in my reasoning about palindromes?

AI Thread Summary
The discussion revolves around the probability of forming palindromes with two and three letters. The original poster mistakenly believes that two-letter palindromes are less likely due to needing the same letter in both positions, resulting in 26 possibilities. However, the reasoning fails to account for the total number of combinations for three-letter words, which includes a middle letter that does not affect the palindrome structure. The key point is that while there are 26 two-letter palindromes, the probability calculations for three-letter palindromes need to consider the total combinations of letters accurately. Understanding the relationship between the number of palindromic combinations and total letter combinations clarifies the error in reasoning.
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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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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The middle letteer doesn't matter (3 letter word). Drop it and get the same as 2 letter word.
 
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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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