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Ch. 4 - Probability
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 4 - ProbabilityProblem 4.4.15
Chapter 4, Problem 4.4.15

Jumble Many newspapers carry “Jumble,” a puzzle in which the reader must unscramble letters to form words. The letters MHRHTY were included in newspapers on the day this exercise was written. How many ways can those letters be arranged? Identify the correct unscrambling, then determine the probability of getting that result by randomly selecting one arrangement of the given letters.

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Step 1: Understand the problem. We are tasked with determining the total number of arrangements of the letters MHRHTY and then calculating the probability of correctly unscrambling the letters to form a specific word. This involves concepts of permutations and probability.
Step 2: Calculate the total number of arrangements of the letters. Since the letters MHRHTY include a repeated letter (H appears twice), the formula for permutations of a multiset is used: \( \frac{n!}{k_1! \cdot k_2! \cdot \ldots \cdot k_r!} \), where \( n \) is the total number of letters, and \( k_1, k_2, \ldots, k_r \) are the frequencies of repeated letters. Here, \( n = 6 \) and \( k_H = 2 \).
Step 3: Apply the formula for permutations. Substitute \( n = 6 \) and \( k_H = 2 \) into the formula: \( \frac{6!}{2!} \). This will give the total number of unique arrangements of the letters.
Step 4: Identify the correct unscrambling of the letters. The correct word formed by unscrambling MHRHTY is 'MYTHHR'. This is one specific arrangement out of the total number of arrangements calculated in Step 3.
Step 5: Calculate the probability of randomly selecting the correct arrangement. The probability is given by \( P = \frac{1}{\text{Total Arrangements}} \). Use the total number of arrangements from Step 3 to compute this probability.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Permutations

Permutations refer to the different ways in which a set of items can be arranged. In this case, the letters MHRHTY can be rearranged in various orders. The formula for calculating permutations of n items, where some items may be identical, is n! / (n1! * n2! * ... * nk!), where n is the total number of items and n1, n2, ..., nk are the counts of each identical item.
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Factorial

A factorial, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics used to calculate permutations and combinations. For example, 5! equals 5 × 4 × 3 × 2 × 1 = 120. Understanding factorials is essential for determining the total arrangements of letters in the puzzle.
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Probability

Probability is the measure of the likelihood that a particular event will occur, expressed as a ratio of favorable outcomes to the total number of possible outcomes. In this context, once the total arrangements of the letters are calculated, the probability of randomly selecting the correct unscrambled word can be found by dividing the number of favorable outcomes (1, for the correct word) by the total arrangements.
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