My mortgage is $1900 a month and I am extremely grateful for that, because it's been that for three years. I chose to buy so that I could have more predictability in my housing costs, and maybe someday I'll move, but then I'll rent it out. All that to say, if you can buying is infinitely better.
At least 10^7 people have refinanced their homes at lower rates, while only 10^5 people have won the lottery. So refinancing is substantially more likely than lottering.