A Two-digit Number Is Chosen At Random. What Is The Probability That The Chosen Number Is A Multiple Of 7?A. \[$ \frac{11}{90} \$\]B. \[$ \frac{13}{90} \$\]C. \[$ \frac{1}{9} \$\]D. \[$ \frac{12}{90} \$\]

by ADMIN 205 views

Introduction

In this article, we will explore the concept of probability and its application in mathematics. We will specifically focus on finding the probability of choosing a two-digit number that is a multiple of 7. This problem requires us to understand the concept of multiples, probability, and the total number of possible outcomes.

Understanding Multiples

A multiple of a number is the product of that number and an integer. For example, the multiples of 7 are 7, 14, 21, 28, and so on. To find the probability of choosing a multiple of 7, we need to first identify all the multiples of 7 within the range of two-digit numbers.

Identifying Multiples of 7

The two-digit numbers range from 10 to 99. To find the multiples of 7 within this range, we can start by listing the multiples of 7 and checking if they fall within the range of two-digit numbers.

  • 7 × 1 = 7 (less than 10, so it's not a two-digit number)
  • 7 × 2 = 14 (within the range of two-digit numbers)
  • 7 × 3 = 21 (within the range of two-digit numbers)
  • 7 × 4 = 28 (within the range of two-digit numbers)
  • 7 × 5 = 35 (within the range of two-digit numbers)
  • 7 × 6 = 42 (within the range of two-digit numbers)
  • 7 × 7 = 49 (within the range of two-digit numbers)
  • 7 × 8 = 56 (within the range of two-digit numbers)
  • 7 × 9 = 63 (within the range of two-digit numbers)
  • 7 × 10 = 70 (within the range of two-digit numbers)
  • 7 × 11 = 77 (within the range of two-digit numbers)
  • 7 × 12 = 84 (within the range of two-digit numbers)
  • 7 × 13 = 91 (within the range of two-digit numbers)
  • 7 × 14 = 98 (within the range of two-digit numbers)

There are 14 multiples of 7 within the range of two-digit numbers.

Calculating Probability

To calculate the probability of choosing a multiple of 7, we need to divide the number of favorable outcomes (multiples of 7) by the total number of possible outcomes (two-digit numbers).

The total number of two-digit numbers is 90 (from 10 to 99).

The number of multiples of 7 is 14.

Probability = Number of favorable outcomes / Total number of possible outcomes = 14 / 90 = 7 / 45

Conclusion

The probability of choosing a two-digit number that is a multiple of 7 is 7/45.

Comparison with Options

Let's compare our calculated probability with the given options:

A. { \frac{11}{90} $}$ B. { \frac{13}{90} $}$ C. { \frac{1}{9} $}$ D. { \frac{12}{90} $}$

Our calculated probability is 7/45, which is not equal to any of the given options.

However, we can simplify the fraction 7/45 by dividing both the numerator and the denominator by their greatest common divisor, which is 1. This gives us the same fraction, 7/45.

Final Answer

The final answer is not among the given options. However, we can express the probability as a fraction: 7/45.

Discussion

This problem requires us to understand the concept of multiples, probability, and the total number of possible outcomes. We can use this problem as a starting point to explore more complex probability problems and to develop our critical thinking skills.

Additional Resources

For more information on probability and its applications, you can refer to the following resources:

  • Khan Academy: Probability
  • MIT OpenCourseWare: Probability and Statistics
  • Wolfram MathWorld: Probability

References

  • [1] "Probability" by Khan Academy
  • [2] "Probability and Statistics" by MIT OpenCourseWare
  • [3] "Probability" by Wolfram MathWorld

Introduction

In our previous article, we explored the concept of probability and its application in mathematics. We specifically focused on finding the probability of choosing a two-digit number that is a multiple of 7. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

Q1: What is the total number of two-digit numbers?

A1: The total number of two-digit numbers is 90 (from 10 to 99).

Q2: How many multiples of 7 are there within the range of two-digit numbers?

A2: There are 14 multiples of 7 within the range of two-digit numbers.

Q3: What is the probability of choosing a multiple of 7?

A3: The probability of choosing a multiple of 7 is 7/45.

Q4: Why is the probability not among the given options?

A4: The probability 7/45 is not among the given options because it cannot be simplified to any of the fractions provided in the options.

Q5: Can you provide more information on probability and its applications?

A5: Yes, we can provide more information on probability and its applications. Probability is a measure of the likelihood of an event occurring. It is used in various fields such as statistics, engineering, economics, and finance. Some of the key concepts in probability include:

  • Random variables: A random variable is a variable that takes on a value based on chance or probability.
  • Probability distributions: A probability distribution is a function that describes the probability of a random variable taking on different values.
  • Expected value: The expected value of a random variable is the average value that it is expected to take on.
  • Variance: The variance of a random variable is a measure of how spread out its values are.

Q6: What are some real-world applications of probability?

A6: Probability has many real-world applications, including:

  • Insurance: Insurance companies use probability to determine the likelihood of an event occurring and to set premiums accordingly.
  • Finance: Financial institutions use probability to model the behavior of financial markets and to make investment decisions.
  • Engineering: Engineers use probability to design and optimize systems, such as bridges and buildings.
  • Medicine: Medical researchers use probability to understand the likelihood of a disease occurring and to develop treatments.

Q7: Can you provide more resources on probability?

A7: Yes, we can provide more resources on probability. Some of the key resources include:

  • Khan Academy: Khan Academy has a comprehensive course on probability that covers the basics and advanced topics.
  • MIT OpenCourseWare: MIT OpenCourseWare has a course on probability and statistics that covers the basics and advanced topics.
  • Wolfram MathWorld: Wolfram MathWorld has a comprehensive article on probability that covers the basics and advanced topics.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide additional information on the topic of probability. We hope that this article has been helpful in understanding the concept of probability and its applications.

Additional Resources

For more information on probability and its applications, you can refer to the following resources:

  • Khan Academy: Probability
  • MIT OpenCourseWare: Probability and Statistics
  • Wolfram MathWorld: Probability

References

  • [1] "Probability" by Khan Academy
  • [2] "Probability and Statistics" by MIT OpenCourseWare
  • [3] "Probability" by Wolfram MathWorld