Why probabilistic syntax for encryption schemes? In Foundations of Cryptography II, Goldreich, the basic mechanism of encryption schemes is written as follows:


  Definition 5.1.1 (encryption scheme): An encryption scheme is a triple, $(G,E,D)$, of probabilistic polynomial-time algorithms satisfying the following two conditions:
  
  
  On input $1^n$, algorithm $G$ (called the key-generator) outputs a pair of bit strings.
  For every pair $(e,d)$ in the range of $G(1^n)$, and for every $\alpha\in\{0,1\}^*$, algorithms $E$ (encryption) and $D$ (decryption) satisfy
  
  Pr$[D(d,E(e,\alpha)) = \alpha] = 1$
  
  
  where the probability is taken over the internal coin tosses of algorithms $E$ and $D$.


I repeated this definition in my own work without too much thought for why the definition was given in probabilistic terms. It was posed to me that it is unnecessary, as written here, for probability to be included and that one could just as well write the consistency equation as $D(d,E(e,\alpha)) = \alpha$. 

I read later in the book that this definition may have the probability bound relaxed to allow for some negligible chance that decryption is not possible, and that this may be necessary for certain public-key encryption schemes.

My question is whether this definition is better, tighter or more complete than one that does not consider the probability of the decryption succeeding? Why is the mechanism of encryption formulated in this way?
Why is the mechanism of encryption formulated in this way?


Writing $D(d,E(e,\alpha)) = \alpha$ would imply that $E(e, \alpha)$ is a unique value, and it generally is not in practice (as most encryption schemes are, in fact, nondeterministic; that is, there are a number of ciphertexts that correspond to any specific plaintext).

The text as written $\text{Pr}[D(d,E(e,\alpha)) = \alpha] = 1$ accounts of the fact that $E(e, \alpha)$ can take on a number of different values (with some probability distribution), and so matches reality (while respecting the formalism).


  I read later in the book that this definition may have the probability bound relaxed to allow for some negligible chance that decryption is not possible, and that this may be necessary for certain public-key encryption schemes.


Yes, it comes up in most lattice-based public key encryption schemes and in some code-based public key encryption schemes.  I have personally never heard of a symmetric system that had the possibility of a decryption failure; that is, where the decryption of the correctly encrypted (and unmodified) ciphertext would not result in the original plaintext.