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Chapter 6: Problem 4

Computers are commonly used to randomly generate digits of telephone numbers to be called when conducting a survey. Can the methods of this section be used to find the probability that when one digit is randomly generated, it is less than 3 ? Why or why not? What is the probability of getting a digit less than 3 ?

Short Answer

Expert verified
Yes, probability is 0.3.

Step by step solution

01

- Understanding the Problem

First, understand that you need to find the probability of randomly generating a digit that is less than 3. Digits range from 0 to 9.
02

- List Valid Digits

Identify the digits that are less than 3. These digits are 0, 1, and 2.
03

- Total Possible Digits

Count the total number of possible digits. The digits range from 0 to 9, which means there are 10 possible digits.
04

- Count Favorable Outcomes

Count the number of favorable outcomes, which are the digits less than 3. There are 3 such digits: 0, 1, and 2.
05

- Calculate Probability

Use the probability formula for this scenario: \[ P(A) = \frac{number\ of\ favourable\ outcomes}{total\ number\ of\ possible\ outcomes} \] So, we have \[ P(digit\ <\ 3) = \frac{3}{10} = 0.3 \]
06

- Conclusion

The methods from this section can be used to find the probability because the problem involves simple random selection from a finite set of equally probable outcomes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Number Generation
Random number generation is the process of producing a sequence of numbers that cannot be predicted reasonably. Computers use algorithms to generate these numbers, mimicking randomness. For surveys, this approach ensures that each digit has an equal chance of being selected. This is especially useful in creating unbiased samples or in simulations. Random digits help avoid patterns and ensure fair representation of the data.
To understand how computers generate random numbers, think of trying to avoid any predictable order. The algorithm makes each digit from 0 to 9 equally likely. This is what we mean by 鈥榬andom鈥�.
Simple Probability
Simple probability measures the likelihood of a single event occurring. Calculating it involves identifying the total number of possible outcomes and the number of favorable outcomes. In our problem, we need to find the probability that a randomly generated digit is less than 3. The total possible digits are 10 (0 through 9).
Among these, the favorable outcomes (digits less than 3) are 0, 1, and 2, totaling 3. The formula for probability is:
\( P(A) = \frac{number\ of\ favourable\ outcomes}{total\ number\ of\ possible\ outcomes} \
P(digit\ <\ 3) = \frac{3}{10} = 0.3 \)
This means a 30% chance of generating a digit less than 3.
Survey Methodology
Survey methodology involves the strategies used to conduct surveys accurately and effectively. When surveying by phone, researchers rely on random number generation to select digits. This avoids bias and ensures all potential respondents are equally likely to be chosen.
Randomly generating phone numbers helps capture a diverse and representative sample of the population. This methodological choice increases the reliability of survey results by minimizing selection bias. Understanding probability helps researchers predict outcomes and improve survey design.
Finite Set
A finite set is a set with a countable number of elements. In our example, the set of digits (0 through 9) is finite. We can easily count all the elements, which total 10.
Finite sets have a limited number of possibilities, making probability calculations straightforward. Each element in our digit set has an equal chance of being chosen, simplifying the process of determining probabilities.
Finite sets are common in everyday problems. Knowing how to work with them is essential in probability and statistics, enabling accurate predictions and analyses.

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Most popular questions from this chapter

Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of \(1 .\) In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places. $$ \text { Greater than }-2.00 $$

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Southwest Airlines currently has a seat width of 17 in. Men have hip breadths that are normally distributed with a mean of \(14.4\) in. and a standard deviation of \(1.0 \mathrm{in}\). (based on anthropometric survey data from Gordon, Churchill, et al.). a. Find the probability that if an individual man is randomly selected, his hip breadth will be greater than 17 in. b. Southwest Airlines uses a Boeing 737 for some of its flights, and that aircraft seats 122 passengers. If the plane is full with 122 randomly selected men, find the probability that these men have a mean hip breadth greater than 17 in. c. Which result should be considered for any changes in seat design: the result from part (a) or part (b)?
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