The letters in the word ARIZONA are arranged randomly. Write your answers in decimal form. Round to the nearest thousandth as needed. What is the probability that the first letter is A? What is the probability that the first letter is Z?] What is the probability that the first letter is a vowel? What is the probability that the first letter is H?
Solution
Step 1 ：The word ARIZONA has 7 letters. The probability of a letter being the first one in a random arrangement is the number of times it appears in the word divided by the total number of letters.
Step 2 ：For the first question, we need to find the probability that the first letter is A. There are 2 A's in ARIZONA, so the probability is \(\frac{2}{7}\) or approximately \(\boxed{0.286}\).
Step 3 ：For the second question, we need to find the probability that the first letter is Z. There is 1 Z in ARIZONA, so the probability is \(\frac{1}{7}\) or approximately \(\boxed{0.143}\).
Step 4 ：For the third question, we need to find the probability that the first letter is a vowel. There are 3 vowels (A, I, O) in ARIZONA, so the probability is \(\frac{3}{7}\) or approximately \(\boxed{0.429}\).
Step 5 ：For the fourth question, we need to find the probability that the first letter is H. There is no H in ARIZONA, so the probability is \(\boxed{0}\).
