How is a ciphertext encrypted using random numbers decrypted using a symmetric key algorithm
Usually the symmetric key is exchanged with Public-key cryptography (which is asymmetric).

Quote from Wikipedia:


  Because of the computational complexity of asymmetric encryption, it is usually used only for small blocks of data, typically the transfer of a symmetric encryption key (e.g. a session key). This symmetric key is then used to encrypt the rest of the potentially long message sequence. The symmetric encryption/decryption is based on simpler algorithms and is much faster.


Concerning random-number-generators in cryptography:

Cryptographically secure pseudorandom number generators are typically used when the design of a cryptographic scheme needs a random number. These include i.e:


One-time pad
Elliptic Curve Digital Signature Algorithm
Cryptographic nonce