A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Concept explainers

Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known …
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosi…
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the prob…
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Question
**Educational Example: Counting Passwords Containing a Specific Digit** **Problem Statement:** Consider the problem "A common type of bike lock requires a six-digit password, and only allows digits 0 through 9. How many passwords are there that contain the digit 9?" **Student's Reasoning:** A student reasons: There are 6 possible places to put the 9, and then no restriction. So there are \(10^5\) possible arrangements of the other digits. This means the correct answer is \(6 \cdot 10^5\). **Multiple Choice Answers:** - **A.** This solution is incorrect, since the student overcounts passwords containing more than one 1. - **B.** This solution is incorrect, since, in the first step, the student needs to specify which digit gets the one. - **C.** This solution is incorrect since there are not \(10 \times 10 \times 10 \times 10 \times 10 \times 10\) arrangements of the digits 0 through 8 in the other three places. - **D.** This solution is correct. **Explanation of the Options:** - **Option A** suggests a misunderstanding about overcounting certain passwords. It assumes a misstep in the counting process, but it doesn't identify a correct counting principle. - **Option B** points out that the student's reasoning might need clarification in identifying which digit gets the 9, which is not relevant in this context. - **Option C** indicates a misunderstanding by claiming there are no arrangements of \(10:10:10:10:10\) for the remaining digits. However, this concept does not correctly analyze the problem. - **Option D** correctly identifies that the student's logic was sound. There are 6 possible positions for the digit 9, and for the remaining positions, there are \(10^5\) possible arrangements of the other digits. This problem illustrates the importance of understanding combinatorial principles in counting distinct possibilities in password formation.
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Transcribed Image Text:**Educational Example: Counting Passwords Containing a Specific Digit** **Problem Statement:** Consider the problem "A common type of bike lock requires a six-digit password, and only allows digits 0 through 9. How many passwords are there that contain the digit 9?" **Student's Reasoning:** A student reasons: There are 6 possible places to put the 9, and then no restriction. So there are \(10^5\) possible arrangements of the other digits. This means the correct answer is \(6 \cdot 10^5\). **Multiple Choice Answers:** - **A.** This solution is incorrect, since the student overcounts passwords containing more than one 1. - **B.** This solution is incorrect, since, in the first step, the student needs to specify which digit gets the one. - **C.** This solution is incorrect since there are not \(10 \times 10 \times 10 \times 10 \times 10 \times 10\) arrangements of the digits 0 through 8 in the other three places. - **D.** This solution is correct. **Explanation of the Options:** - **Option A** suggests a misunderstanding about overcounting certain passwords. It assumes a misstep in the counting process, but it doesn't identify a correct counting principle. - **Option B** points out that the student's reasoning might need clarification in identifying which digit gets the 9, which is not relevant in this context. - **Option C** indicates a misunderstanding by claiming there are no arrangements of \(10:10:10:10:10\) for the remaining digits. However, this concept does not correctly analyze the problem. - **Option D** correctly identifies that the student's logic was sound. There are 6 possible positions for the digit 9, and for the remaining positions, there are \(10^5\) possible arrangements of the other digits. This problem illustrates the importance of understanding combinatorial principles in counting distinct possibilities in password formation.
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