You are right: every form of contraception does have a failure rate. You are not going to find a 100% effectiveness rate (even vasectomy is not 100% effective in the first 3-4 months).
Luckily, the statistics do not increase every year as you suggested; there is not an "multiplying effect" on the numbers (I forget the correct statistical terms). Your worry of: "if 1% chance over a year equals 20% over 20 years" is not statistically correct. Every year, you have less than 1% chance of becoming pregnant when using one of the most effective forms of contraception consistently and correctly.
Have you heard of emergency contraception? That may ease your fears. If you are using a condom and it breaks, or forget to take a birth control pill, there is something you can take up to 72 hours after intercourse to help prevent a pregnancy. You can read more about emergency contraception.
Here is a chart that compares the effectiveness rates of various types of birth control methods.
You can learn more about all of your options at How birth control works at the ACOG site.
Something that might also ease your fears: you might want to read through the above brochures and learn about your fear: how do women get pregnant? It's actually not as easy as you may think! A woman has only a 20% chance of becoming pregnant in any one menstrual cycle.
You can also learn about when you ovulate, and know that you have control about some of the mitigating factors (since we're talking about statistics). For instance, you can increase your odds by: not having intercourse during ovulation, you can use a form of hormonal contraception plus a condom.
hope this helps!