Casework and Law of Total Probability

A shoe store orders all of its shoes from two factories.

• Factory A makes $47\%$ of the shoes sold at the shoe store.
• Factory B makes $53\%$ of the shoes sold at the shoe store.

Occasionally, each factory incorrectly sizes their shoes and instead of getting a pair of size nine shoes you might get two different sizes.

• Factory A is guilty of doing this $22\%$ of the time.
• Factory B only makes this mistake $15\%$ of the time.

If Sarah goes to the store and buys a pair of shoes, what are the chances that she ends up with a pair of shoes that are incorrectly sized?