You start out with the string "ABCDE".

Every minute you randomly knock one letter off of the beginning or the end.

What is the probability that when there is one letter left it will be a C?

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Let's name the two operations, say $F=$ removing front character and $R=$ removing rear character.

So, total number of ways to keep last character is done by four operations.So number of ways is $2^4=16$ .

Now to keep last character as $C$ the following operations can be done

$F \to R \to F \to R \\ F \to F \to R \to R \\ F \to R \to R \to F$

Now $F$ can be replaced by $R$ and we get again $3$ methods.

So, total ways $3+3=6$

So, the required probability is $\dfrac{6}{16}=\dfrac{3}{8}$