Why aren't rainbows more common? Why are rainbows relatively rare?  On any given day, there are billions of water drops in the air of varying sizes and dispersions, all of which light is passing through and refracting.  What physical phenomenon has to occur so that these drops combine to form a rainbow?  Why is it more digital (visible or not) than analog?  Is it that the drops have to be consistently dispersed and sized?  If that was the case, why don't rainbows appear just in the regions that have a suitable makeup?  Or does a single drop cause the refraction we see across the sky?  I was trying to explain rainbows to my kids but got totally confused...
Here is an example of a rainbow at my work, 7:00am in Edwards, California, facing west, no rain, but it was partly cloudy.  The Rainbow seemed to have no connection to the clouds.  
Hopefully the answers will allow for the existence of this rainbow.  Thanks!


 A: There are many obvious reasons why rainbows are not very common or, more precisely, are not very commonly observed by any particular individual: it has to be rain, it has to be sun, it has to be this or that part of the day, East or West, dark background, etc. and that individual has to be in the right spot at the right time and actually look. 
Since we tend to be indoors during rain, we tend to see rainbows, when we travel and get caught by rain in the middle of the road, without tall buildings or trees blocking the view.
So, if all those other things line up, will we always see a rainbow or does it have to be a particular type of rain or particular type of water droplets? Here are couple of additional factors affecting the probability/visibility of a rainbow:
Droplet size and shape. For the brightest rainbow, droplets have to be 1-2mm. Droplets smaller than 0.05mm will create a white rainbow, etc. More details could be found here.
Droplets density. For the rainbow to work, a photon has to hit one droplet and fly right back to your eye, i.e., it has to be scattered once. If the density of droplets is high, like in clouds, a photon will likely hit multiple droplets and end up moving at a random angle, breaking the 42 degrees requirement.
A: There are no billions of water drops in the air on any given day. Water in the air is normally in the form of vapor, not drops. For drops to form, the relative humidity should be 100% causing condensation, such as during or after a rain. 
The second requirement is a direct sunlight from behind. You can easily create a rainbow on a sunny day while watering your lawn. Stand with your back to the sun and spray (or better mist) water widely in front of you. You should see a rainbow centered around the shadow of your head. And when you see a real rainbow after a rain, notice that it is also centered around the spot where the shadow of your head is or would be at the moment. This typically implies that the sun should be fairly low in the sky (e.g. it is less likely to see a rainbow at noon when the sun is high). 
It is also possible, although rather rare, to see a rainbow from an airplane. I have seen it only once in over 200 flights. It was in multiple full circles around the shadow of the plane when it was passing over the clouds. 
A: As the other answers said, there are a few necessary conditions so you can see a rainbow. Perhaps the most crucial here is light coming from a single direction, usually the sun. It may not be completely intuitive why this is necessary: if light from one direction forms one rainbow, won't light from multiple directions just form even more rainbows?
Well, yes. Just...

When multiple rainbows overlap, they tend to do so “destructively”: the separated colour components merge again. The limit of this is the everything looks white/grey, which is of course exactly what we se when immersed in fog, heavy rain or in a cloud.
A: A number of conditions have to be just right in order to see a rainbow.


*

*The Sun has to be visible in the sky. Rainbows don't occur on overcast days. The light hitting the raindrops needs to come from what is close to a point source to have the reflections and refractions in a myriad number of raindrops combine to form a rainbow. Diffuse light (overcast conditions): No rainbow.

*The Sun has to be fairly low to the horizon. The primary bow forms a cone with your eye as the vertex and the line from the Sun through your head as the axis, with the red light at an angle of about 42° from the axis and the blue,  about 40°. The Sun needs to be below 40° above the horizon to see a rainbow, and at that high of an angle, the rainbow won't be very good. Rainbows are best in when they form less than an hour or so after sunrise or less than an hour or so before sunset.

*Rain needs to be falling opposite the Sun. Off to the side: No rainbow. From a very low cloud at the horizon: No rainbow.

*It has to be rain rather than a fog or a mist. Cloud droplets are far too small to form a rainbow. Cloud droplets are about the same size as the wavelength of visible light. This means light hitting cloud droplets is diffracted rather than reflected and refracted. Clouds form glories, coronae, and fogbows. The latter are similar to rainbows, but without color. Mists form at best fuzzy rainbows; the drops are small that diffraction dominates over reflection and refraction. The drops need to be about a millimeter in diameter to form a rainbow.


Altogether, this makes rainbows rather rare.
A: The drop size has a direct correlation to the saturation of the colors in the rainbow. You can see examples here, notice how the rainbow is much brighter when the drops are larger, and the fades as they become smaller in the mist.
The "drops" in the air around you are normally microscopic, and as a result, the rainbow you would see if you looked in the right direction simply isn't visible. You might see one around the sun or through some clouds, but generally its simply not strong enough to overpower the strong blue (or grey) background.
You also need to remember that you can only see it when the sun is low in the sky, so for most of the day they aren't visible anyway.
