Analogue television using the PAL system requires a transmitted reference burst to operate a 4.33 MHz oscillator. If this is missing the colour decoder will not operate and a colour killer circuit operates to avoid colour noise on a monochrome transmission.

On a digital transmission there will be no snow just a blank screen as there is no useable signal for the digital decoder.

The short answer is: the analogue colour decoder requires a 10 cycle burst of the colour carrier at the start of each line to operate. This applies to the PAL, NTSC and SECAM coding schemes.

As a schoolboy, I marvelled at the ingenuity of analogue colour tv which coded the colour information in a fine moving dot pattern added to the monochrome picture. On a black & white tv one could see it if one looked closely.  The colour decoder takes this pattern, extracts the colouring signal as synchronized by the 10 cycle burst, and "tints" the monochrome picture. The dot pattern is designed not to clash with normal picture elements, but accidents do happen.  The most common artifact [in the PAL system] is fine stripy red-green banding at some sharp edges.  One time, however, I saw a Randolph Scott Western where the hero wore a finely checkered shirt.  At a certain distance from the camera, the shirt would suddenly go bright purple.

Pure white noise would not trigger any colour decoding.

In the PAL colour TV system the decoder must see the ten cycles of a very specific frequency of 4.43361875 Hertz(cycles per second in the old system). If this is not present the "colour killer" circuit will detect that there is nothing there and force the system to display in monochrome or "black-and-white". Because "noise" doesn't have this very particular arrangement of signals, just a jumble, it will never get detected as a valid colour signal. In theory anyway. Sometimes, in practice, you will see coloured noise fleetingly.

In my previous reply I stated "4.43361875 Hertz", It should in fact be 4.43361875 MegaHertz. Sorry about that slip-up. This was referring to the "burst" or colour sub-carrier frequency in a analogue PAL TV colour signal.

In layman's terms, in order for the set to display color it must receive a signal that contains color reference information that the tv is expecting to see. 

In the absence of the signal (called the color burst others here mention, which occurs just after the sync pulse that starts each horizontal scan line), the circuitry of the tv is designed to remain compatible with the older black-and-white standards, and therefore shows random noise as black-and-white (called luminance information).