1. **State the problem:** We want to find the probability that a randomly chosen customer purchased either popcorn or a hamburger. 2. **Identify the data:** From the table, the purchases are: - Hamburger: French Fries 83, Peanuts 2, Popcorn 19 - Pizza: French Fries 67, Peanuts 5, Popcorn 29 - Deli Sandwich: French Fries 37, Peanuts 14, Popcorn 3 3. **Calculate total customers:** Sum all purchases: $$\text{Total} = 83 + 67 + 37 + 2 + 5 + 14 + 19 + 29 + 3 = 259$$ 4. **Calculate number who purchased hamburger:** Sum all hamburger purchases: $$\text{Hamburger total} = 83 + 2 + 19 = 104$$ 5. **Calculate number who purchased popcorn:** Sum all popcorn purchases: $$\text{Popcorn total} = 19 + 29 + 3 = 51$$ 6. **Calculate number who purchased both hamburger and popcorn:** The overlap is the number who purchased popcorn and hamburger together, which is the popcorn under hamburger category: $$\text{Popcorn and Hamburger} = 19$$ 7. **Use the formula for union of two events:** $$P(\text{Popcorn or Hamburger}) = \frac{\text{Popcorn} + \text{Hamburger} - \text{Both}}{\text{Total}}$$ 8. **Substitute values:** $$P = \frac{51 + 104 - 19}{259} = \frac{136}{259}$$ 9. **Simplify the fraction:** Find gcd of 136 and 259. - Factors of 136: 1, 2, 4, 8, 17, 34, 68, 136 - Factors of 259: 1, 7, 37, 259 No common factors except 1, so fraction is already in simplest form. **Final answer:** $$P(\text{Popcorn or Hamburger}) = \frac{136}{259}$$