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Average Price for Home Appraisal - Professionaloutdoormedia

📅 May 14, 2026 👤 scraface

May 14, 2026

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📌 Question: A historian of science is studying the development of probability theory and examines a 17th-century manuscript containing all permutations of the letters in the word PROBABILITY. If the historian randomly selects one such permutation, what is the probability that the vowels appear in alphabetical order (i.e., A, I, O — but in this word, only A and I and O appear; note: PROBABILITY has letters: P, R, O, B, A, B, I, L, T, Y — vowels are A, O, I. Wait: check vowels: A, O, I. So vowels are A, I, O — and the condition is that in the permutation, A, I, O appear in alphabetical order: A before I before O.

📌 Vowels in PROBABILITY are A, I, O (since B, R, B, L, T, Y are consonants). So we have three distinct vowels: A, I, O.

📌 Total number of distinct permutations of the letters in PROBABILITY:

📌 Repeated letters: two Bs, and rest are unique

📌 Now, we want the probability that in a random permutation, the vowels A, I, O appear in alphabetical order (A before I before O), regardless of position among other letters.

📌 Is this valid? Yes: since the non-vowel letters are symmetric and do not affect relative order, and all positions are equally likely, the relative order of the three distinct vowels A, I, O is uniformly distributed over all 6 permutations. Hence, the probability they appear in alphabetical order is:

📌 But wait: are the vowels always distinct and clearly identifiable? Yes — A, I, O are all distinct and appear once.

📌 Question: Jackson rolls three fair 8-sided dice, each numbered from 1 to 8. What is the probability that exactly two of the dice show a prime number?

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