What are the chances of out-growing a food allergy if you started getting your allergies when you were a child?

Depends. The most common food allergies are egg, milk, wheat, soy, peanut, tree nuts and fish/shell fish. Most will outgrow egg, milk, wheat and soy. Some will outgrow peanut and tree nut. Shellfish not too many outgrow.
Possible. Many children who had food allergies outgrow them with time. But it varies with the type of food. Kids that are allergic to soy, eggs and cow´s milk mostly outgrow the problem by their teen years. The story is different for tree nuts and peanuts, these usually persist.