I hate to tell you this, but the chance of getting flats has nothing to do with whether it's a carbon fiber bike or not. It's all about how whether it's raining and hard it is going to be to be to get the tire off the rim.

My (dismal) record is 4 tubes on a single flat. One tube had a faulty stem, one I pinch-flatted, one the good Samaritan who stopped to lend me his spare tube pinch flatted and the final one was the original, nicely patched.

That one is still on the bike, I think.