The short and academically correct answer: a random noise signal is random. The lack of organization precludes that there is no separate color carrier present, and the set goes, as is already answered, into b/w mode. For split seconds there might be randomly something that looks like a color carrier might be present, then there might be a quick colorish flicker.

yes, you do not see wrong:

while the generated signal is strictly b/w there are reasons that the academical requirement for b/w noise is not really happening: technical reasons and physiological reasons.

A little ramble of why we still see sometimes color in real random noise:

There are:

physiological reasons: 'optical illusion'

reasons that have to do with improper function of an analog TV

reasons that have to do with improper function of analog to digital conversion that can make random noise appear tinted on a digital TV (a 'should not happen' that happens actually all the time on cheap TVs, there is a reason for 'cheap' and 'costly' digital TVs...)

Let's go, or just stop here if you like... everything has basically been said, this is just for your enjoyment:

pink artifacts:

'snow', when we see it: our eyes do add color tints , as they do add 'correlations'. Some see faces and stuff (TV people...maybe talk to someone about it... ;-D )

there are some optical illusions out there that move some b/w patterns and we (at least I do, don't have my bionic eyes yet..) do see color although there is none at all on the moving pattern google for 'optical illusions'

Technical reasons why 'should be b/w noise' may have some hues and colored pixels:

color in random noise on an analog color TV (or computer monitor):

Reminding: noise from 'no signal' or from a blank analog magnetic medium on an analog TV cannot have a color signal which would require the organization of the noise into frames of certain durations ...

There are instances where a color TV will render random noise (b/w) as color:

The color convergence is out of alignment and the three (r/b/g) guns focus slightly off-center of the color triple on the screen (instead of 100% r/b/g it gives a little tint), either always, or per pixel and randomly. Stationary tints mean you need to demagnetize, random tints per pixel mean there is noise on the beam control.

"I see color noise on a digital TV..."

A digital TV has no noise, it works or not.

A rendered analog signal (from a VCR empty tape or from an analog tuner / receiver) will be b/w at the origin.

Then the analog signal is converted to a digital signal and here trouble can lurk in the detail:

Before the conversion from analog to digital, the above rules for analog signals do apply: no blanking period means no color signal went in.

Your analog inputs to a digital TV or monitor can come from

dual tuners:

even if there are no analog TV stations you might have an analog signal from some arcade games, from a camera abusing of an analog TV channel, or an old VCR you light into your digital TV via analog channel 2 or 3 (you need to know what a rotary dial phone is to understand this....).

Again: academically these should be strictly b/w after conversion from analog to digital.

Some reasons how conversion imprecision can add color to 'white' noise:

Bad converters used to convert from analog to digital can include 'dust' in the rapidly generated digital pixel value.

That is a software issue in the converter. A 0 is not exactly zero, a one is a little less than all bits on, and some bits are picked sometimes in the middle. When I say here and from here on '0' and '1' I am talking about 'all bits 0' and 'all bits 1' in the per pixel color depth value.

Bad video a/d converter designs use FP units on fast and cheap general-purpose chips: again: 0 is not exactly zero ever and 1 is at most .999999. they do some mantissa tricking to process, e.g., a 24 bit pixel signal through an FP unit which is a general-purpose, cheaper (and faster, vectorized) chip than a real video converter chip.

This creates a hue that is different per pixel and per instance (the same digital pixel lights up in slightly different hues each time it is used).