Speaking from a place of white privilege, I don't know that I can even answer authoritatively. I will say that in the conversations I've had with those less fortunate than myself, the discussion usually turns to "don't feel bad for your opportunities, but don't downplay their importance, either." Ex-girlfriend's little sis went to the same private school as Bill Gates. She had a friend whose dad was the CEO of Alaska Airlines (having been ousted from Northwest for fucking it up badly enough for Delta to eat it years later). That kid? Every bit as privileged as you might imagine.
Trying to parse all the things that count as white privilege is difficult. Opportunity is the biggest component, but I don't think it's the only one. I once got pulled over making a right on a no turn on red. From the left hand turn lane. At 2:00 am. After just having left the bar. Where I had drank probably 8 beers (mas o menos). And I couldn't find my registration. Or my proof of insurance. What happened next? The cop gave me directions, and sent me on my way. Seriously. True story. Maybe he was not very good at his job, or maybe I was a very good actor, but either way the default position is to trust me. I don't think a black guy in that circumstance would've received the benefit of the doubt. It's not just unequal opportunity, but also unequal protection that defines whiteness, I guess is my point.
I had an experience earlier this year where if I had been a black kid, I would have undoubtedly, absolutely, no-question been detained, searched, arrested and booked. Instead the cop told me to go on my way and avoid driving by the same location on my way home (I was heading out and he knew I'd be coming back). I think about that a lot. I think it's white privilege + pretty, tiny female. I was thinking about what if I had insisted I be arrested/treated fairly instead...which would probably have cost me my job, my license, maybe a few other things besides. It sucks. The person I am thinks I should have insisted I be prosecuted and treated - but the costs weren't worth it - but the justice system is fucked up. I take advantage of who I am and where I was placed in life when I was born into it and I can't help it in a lot of ways. It sucks. But to not take advantage of these things is to also allow life to fuck me up and fail me. They say it don't be like it is, but it do.
This discussion (actually any discussion of white privilege, really) reminds me of this piece by Christopher Guest, from 1969 :