How high are the chances of having breast cancer and cervical cancer at the age of 24?

Very low: For most women, so low as to be nearly zero. However, there are familial cancers that affect the breast and ovaries (rather than cervix/uterus), and in those families cancer can strike in the 20s.

Not THAT low...: Cancer of the cervix, or at least precancer picked up on a pap smear, is actually quite common in women in their 20's. And breast cancer, especially in its preinvasive form, isn't a rarity either. It may be dumb luck, or a cancer family syndrome such as li-fraumeni may be operating. Thankfully, the large majority of both types of cancer are cured, especially if caught early.

Mixed picture: Outside of genetic causes, breast cancer in a 24 year old is very uncommon, although it can occur. Cervical cancer, on the other hand is not uncommon in that age group, and would be much more of a concern. Fortunately, having regular pap smears is an excellent way of screening for the precursors of cervical cancer and should be part of your healthcare plan.