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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- 1. Intro to Stats and Collecting Data1h 14m
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- 5. Binomial Distribution & Discrete Random Variables3h 6m
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- 13. Chi-Square Tests & Goodness of Fit2h 21m
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4. Probability
Fundamental Counting Principle
Problem 3.1.66a
Textbook Question
66. Access Code An access code consists of six characters. For each character, any letter or number can be used, with the exceptions that the first character cannot be 0 and the last two characters must be odd numbers.
a. What is the probability of randomly selecting the correct access code on the first try?
Verified step by step guidance1
Step 1: Determine the total number of possible characters for each position in the access code. There are 36 possible characters (26 letters + 10 digits) for most positions, except for the first and last two positions which have specific restrictions.
Step 2: For the first character, it cannot be '0', so there are 35 possible choices (26 letters + 9 digits excluding '0').
Step 3: For the last two characters, they must be odd numbers. There are 5 odd digits (1, 3, 5, 7, 9), so each of these two positions has 5 possible choices.
Step 4: For the remaining three middle characters, there are no restrictions, so each of these positions has 36 possible choices.
Step 5: Calculate the total number of possible access codes by multiplying the number of choices for each position: (35 choices for the first character) × (36 choices for the second character) × (36 choices for the third character) × (36 choices for the fourth character) × (5 choices for the fifth character) × (5 choices for the sixth character). Then, calculate the probability of randomly selecting the correct access code on the first try by taking the reciprocal of this total number of possible access codes.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Combinatorics
Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. In this context, it helps determine the total number of possible access codes by considering the restrictions on character selection, such as the first character not being '0' and the last two characters being odd numbers.
Probability
Probability is a measure of the likelihood that an event will occur, expressed as a ratio of favorable outcomes to the total number of possible outcomes. To find the probability of selecting the correct access code on the first try, one must calculate the ratio of successful outcomes (1 correct code) to the total number of valid access codes.
Recommended video:
Introduction to Probability
Permutations and Restrictions
Permutations refer to the different ways in which a set of items can be arranged. In this scenario, the restrictions on the first character and the last two characters create specific conditions that must be accounted for when calculating the total number of valid access codes, impacting the overall probability.
Recommended video:
Introduction to Permutations
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