Foot cut through deep, and bone broken. What's the chance of bone infection after it's stitched up?

Can happen. Any time the skin is broken, bacteria can get in. Infection is more common in diabetics and people with vascular disease, though it happens to healthy people also. Your doctor will probably give antibiotics and a tetanus shot depending on the nature of the cut and break as well as cleaning instructions. If you use tobacco, please stop if at all possible.
There is chance. The chance of infection there but we could minimize it by: clean the wound truly and use antibiotic after that.
High... If you can see the broken bone, then your chances of having it contaminated with bacteria is high. Probably best to assume that it is infected. I often take cultures of the bone just to prepare for the possibility.
OPEN FRACTURE. You've had an open fracture. There are three grades; III worse than II, II worse than I. The infection rate increases as you go from I to II to III. Infection can occur in bone or surrounding tissue. Watch for redness, swelling, increased pain. Keep your appointments with your doctor! Sincerely, Dr. Latva.