I think Tzva means tubes not tires in your seat bag.

I ride a lot and I just have 2 extra tubes, a multipurpose bike tool, and 3 tire irons (plastic) that fit together in my under seat bag. Be sure to cover the tubes with (REAL) talcum powder before you stash them in the bag. It will make changing flats a whole lot easier!

I use a Road Morph pump with a folding step and pressure gauge. That way you pump as you would at home, going down instead of sideways, to get air in the tire. That pump is fastened onto the frame of my bike and strapped on with velcro to keep it secure.

I also have a "bento box" fastened to my top tube thanks to the TE girls a few years ago who raved about this little goodie! I can put my cell phone in there as well as my cleat covers and paper towel for messy jobs like flat tires on the back wheel. Wallet with identification and extra food go in back pockets.

The biggest problem I"ve had in the 5-6 years I've ridden have been flat tires. Yes, you should have something to temporarily fix a tire should you ever have a blowout (dollar bill works fine). Hopefully you'll be riding with others who can help you out should anything happen on the road.