High School Where is the error in my reasoning about palindromes?

[red65]

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!

 

[PeroK]

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.

 

[mathman]

The middle letteer doesn't matter (3 letter word). Drop it and get the same as 2 letter word.

 

[pbuk]

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?