Ice axe *or* crampons?

If you were going on a winter walk where some snow/ice was likely, but
nothing particularly difficult or dangerous, which would you take
first, and why? (Note the emphasis on walk - I don't mean
front-pointing up gullies.)

I used to think the decision was considered straightforward, but I find
that the majority of people I encounter in the hills (round here at
least - which is Japan) make the opposing decision.

James

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