Question

Mr Balaram picked a prime number between the integers 1 and \(20 .\) What is the probability that it will be number 13 ? (1) \(\frac{7}{8}\) (2) \(\frac{1}{20}\) (3) \(\frac{1}{8}\) (4) \(\frac{13}{20}\)

Short answer

Answer: (3) \(\frac{1}{8}\)

Step-by-step solution

Find the prime numbers between 1 and 20

A prime number is a number greater than 1 whose only factors are 1 and itself. The prime numbers between 1 and 20 are: 2, 3, 5, 7, 11, 13, 17, and 19. There are 8 prime numbers in this range.

Calculate the probability of picking the number 13

Since there are 8 prime numbers between 1 and 20, the probability of randomly picking the number 13 from these prime numbers can be represented as a fraction where the numerator is 1 (since 13 is only one of the prime numbers) and the denominator is the total number of prime numbers (which is 8). Therefore, the probability of picking the number 13 is \(\frac{1}{8}\).

Find the answer in the given options

The calculated probability, \(\frac{1}{8}\), matches option (3). So the correct answer is (3) \(\frac{1}{8}\).

Key concepts

Prime Numbers

Prime numbers are fascinating numbers in mathematics. They are numbers greater than 1 that have no divisors other than 1 and themselves. In simple terms, they are numbers that cannot be divided evenly by any number other than 1 and the number itself. This makes them the building blocks of all natural numbers.

Prime numbers are important for various reasons. They are crucial in number theory and have applications in fields like cryptography. For example, secure communication over the internet often relies on the properties of prime numbers.

In the exercise, we are interested in identifying prime numbers between 1 and 20. The primes in this range are 2, 3, 5, 7, 11, 13, 17, and 19. Notice how each of these numbers cannot be divided any further, except by 1 or their own value. Understanding which numbers are prime helps in various mathematical problems, including calculating probabilities like we did in the exercise.

Fractions

Fractions are used to represent parts of a whole. They are written in the form of two numbers separated by a slash, such as \(\frac{1}{8}\). The top number is called the numerator, and it signifies the number of parts we are considering. The bottom number is called the denominator, and it tells us the total number of equal parts the whole is divided into.

In probability, fractions are used to express the likelihood of an event occurring. In our example, the probability of selecting the number 13 from the prime numbers between 1 and 20 is \(\frac{1}{8}\). This fraction indicates that there is 1 favorable outcome (choosing 13) out of 8 possible outcomes (all the prime numbers). By using fractions, we simplify complex likelihoods into an easily understandable form.

Fractions not only represent probabilities but are used widely in everyday life for measurements, dividing resources, and much more. Understanding fractions is key to mastering various areas of mathematics.

Numerator and Denominator

In any fraction, the two crucial components are the numerator and the denominator. These two elements define the fraction's value and meaning.

The **numerator** is the number above the line in a fraction. It represents how many parts we consider or focus on from a whole set. For instance, in the fraction \(\frac{1}{8}\), ‘1’ is the numerator, signifying one single part that we are looking at.

The **denominator** is the number below the line. It denotes the total number of equal parts into which the whole is divided. In our fraction \(\frac{1}{8}\), ‘8’ is the denominator. It tells us that there are eight parts in total.

Understanding these components helps us decipher any fraction and thus comprehend essential concepts in probability and mathematics. When working with fractions, remember that both the numerator and denominator play critical roles in determining the fraction's actual value.