How fast does your chance of getting cancer go down after quitting smoking?

Not fast enough. If some one quits after several years of smoking, due to yeasrs of exposure still they can get lung cancer but incidence may be slighly less than some one still smoking. Only conclusion is never smoke, dont get exposed to secondary smoke.
Relatively fast. Exact figures that are accurate are not easy to come by, as studies are done on limited populations, but, for example, the national cancer institute followed 104000 women over more than 20 years, and found that after just 5 years of smoking cessation, the risk of gettting lung cancer dropped by 21% while the risk of heart disease dropped by 50% versus those who kept smoking.