I'll second WD on this.

Sometimes, even if you do everything right, you won't be able to stop yourself. I have done a lot of snow and ice climbing over the years and am reasonably practiced at self-arrest. Once, while climbing Boston Peak in the North Cascades, I was cutting steps across a stretch of water-ice on a 45 degree slope when one of my steps flaked off; I was not wearing crampons. They wouldn't have helped here and, in fact, they could have made things worse.

I quickly got into arrest position (toes and pick)and then just rode the ice axe pick, at high speed, down the hard ice slope into a pile of boulders. Fortunately (read incredibly) the only damage to me was a magnificent set of bruises on my legs and butt. I was able to walk out, albeit, slowly.

I came away from this experience with both a renewed respect for how quickly things can happen and a great appreciation for putting in an anchor and a belay before it is needed rather than after.

My companion on this climb was on softer snow and was using a casual boot/axe belay. Moreover,he was wool-gathering (sight-seeing) when I fell. The first he knew about it was when he was yanked out of stance and was sliding down-slope himself. He was able to self-arrest; I think the force expended on yanking him from his stance slowed me enough to avoid worse injury.

The margin for error on ice and snow is quite narrow. Experience and proper equipment can widen the margin but the risk is always there. It is up to the individual to judge how far to push things. But, such judgment can only come with experience. One should never be cocky or cavalier (as I was) on any snow or ice slope with a cliff, or a pile of rocks, at the bottom.

Marginal equipment and lack of knowledge of how to use it are a potentially deadly combination.