Where is the error in my reasoning about palindromes?

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Discussion Overview

The discussion revolves around the probability of forming palindromes with 2 and 3 letters. Participants are examining the reasoning behind the likelihood of creating such palindromes and identifying potential errors in the initial reasoning presented.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that forming a 2-letter palindrome is less likely because it requires the same letter in both positions, resulting in 26 possibilities.
  • Another participant argues that for 3-letter palindromes, there are 26 possibilities for the first and last letters and 26 for the middle letter, leading to a total of 676 combinations.
  • A later reply points out that not all possibilities are equally likely, noting that a specific palindrome (like "aa") is 26 times more likely than a 3-letter palindrome (like "aaa").
  • Another participant states that the middle letter in a 3-letter palindrome does not affect the count, suggesting that it can be disregarded to compare it to a 2-letter palindrome.
  • One participant questions the denominator in the probability calculation for 2-letter palindromes, indicating that clarity is needed on what constitutes the total number of outcomes (Y).

Areas of Agreement / Disagreement

Participants express differing views on the reasoning behind the probabilities of palindromes, and the discussion remains unresolved regarding the correct interpretation of the probabilities involved.

Contextual Notes

There are unresolved questions about the total number of possible outcomes (Y) in the probability calculations, and the implications of letter positions in determining palindrome likelihoods are not fully clarified.

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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