I was having a problem like this with my laptop a few weeks ago and it was driving me up the walls.
It turned out to be a new driver for my trackpad that added multi-touch support. It had a setting where when I tapped with three fingers it would open bing.com by default. So sometimes I would just hit it by chance and fucking bing would pop up. I thought microsoft was fucking with me and trying to boost their visitation numbers but nope, just synaptics being weird. So... something to check
I forgot to mention that when I accessed the recovery partition the first time and before uninstalling MyWaySA on my own the second time, I did a Malwarebytes scan. MyWaySA was detected. Someone noticed the same thing too. [link]
If that's not the answer you're looking for, it's probable that the Bing ad is a pop-up from the microsoft site or some other site. If this happens on random sites, it could be some kind of malicious add-on. There are some known microsoft-made malware which redirect you to bing, if I recall correctly. Although I think those are probably toolbars.
Anyway, to prevent future misfortune, get these addons for firefox:
AdBlock Plus NoScript DNT+
Of course, you could always install linux and not worry about all this malware biz