1. **Problem statement:** The letters in the word ARIZONA are arranged randomly. We want to find the probability that the first letter is Z. 2. **Total letters:** The word ARIZONA has 7 letters: A, R, I, Z, O, N, A. 3. **Total possible first letters:** Since the letters are arranged randomly, each letter has an equal chance to be first. There are 7 letters, so 7 possible first letters. 4. **Number of Z's:** There is exactly 1 Z in ARIZONA. 5. **Probability formula:** Probability = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}. 6. **Calculate probability:** $$ \text{Probability(first letter is Z)} = \frac{1}{7} \approx 0.143 $$ **Final answer:** The probability that the first letter is Z is approximately 0.143.