Question

20 cards are numbered 1 to \(20 .\) If a card is selected at random, then what is the probability that the selected number will be an odd prime? (1) \(2 / 5\) (2) \(3 / 5\) (3) \(3 / 10\) (4) \(7 / 20\)

Short answer

Answer: The probability is \(7/20\).

Step-by-step solution

Understanding odd prime numbers

Prime numbers are those numbers that are only divisible by 1 and themselves. Odd prime numbers are essentially prime numbers that are also odd. All prime numbers are odd, except for the number 2.

Identifying the odd prime numbers

List all the prime numbers from 1 to 20 and identify the odd ones: 2, 3, 5, 7, 11, 13, 17, and 19. Since 2 is the only even prime, the odd prime numbers in our range are 3, 5, 7, 11, 13, 17, and 19.

Counting the odd prime numbers

We have already determined that there are 7 odd prime numbers: 3, 5, 7, 11, 13, 17, and 19.

Calculating the probability

Now we need to calculate the probability of selecting one of these odd prime numbers from the 20 cards. The probability is calculated by dividing the number of odd prime numbers (7) by the total number of cards (20). Thus, the probability of selecting an odd prime number is \(\frac{7}{20}\). Therefore, the correct answer is (4) \(7/20\).

Key concepts

Prime Numbers

Exploring the world of numbers, prime numbers stand out for their unique properties. These are the numbers greater than 1 that have no positive divisors other than 1 and themselves. Think of them as the 'building blocks' of the whole numbers, because every number can be factored down to a unique set of prime numbers.

For instance, the prime numbers between 1 and 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that all of them, with the sole exception of 2, are odd numbers. This makes 2 the very special 'even prime number'. The concept plays a vital role in solving our exercise, as the recognition of these foundational numbers helps in determining the result of our probability calculation, especially when looking for the odd ones among them.

Importance of Prime Recognition

A strong understanding of primes is essential for various mathematical concepts including number theory and cryptography. Recognizing primes can also serve as a helpful step in simplifying fractions, finding least common multiples, and even solving puzzles and brainteasers that involve divisibility.

Odds and Evens Mathematics

Stepping into the realm of basic number classification, we encounter the concept of odd and even numbers. An even number is any integer that can be exactly divided by 2, resulting in another integer. On the flip side, an odd number is an integer that cannot be evenly divided by 2.

Odd numbers end in 1, 3, 5, 7, or 9, while even numbers terminate in 0, 2, 4, 6, or 8 when expressed in the decimal system. This division of numbers into odd and even is a fundamental part of arithmetic and offers a simple way to understand number properties, patterns, and relationships.

Applying Odds and Evens

In our exercise, understanding the difference between odd and even is crucial, because it directly influences the pool of numbers we consider for our probability calculation. For instance, when selecting an odd prime as the exercise requires, we immediately disregard any even numbers, saving time and streamlining our approach.

Basic Probability

Probability is the likelihood or chance of a particular event occurring. It's a fundamental concept used in various fields such as mathematics, statistics, finance, gambling, and even daily decision making. Probability is usually expressed as a fraction or a decimal, ranging from 0 to 1, where 0 indicates impossibility and 1 denotes certainty.

To determine the probability of a single event, you divide the number of favorable outcomes by the total number of possible outcomes. For example, if you're drawing cards labeled 1 to 20 and you want to draw an odd prime, the probability is the count of odd prime numbers (favorable outcomes) divided by 20 (total outcomes), which, as per our exercise, yields \(\frac{7}{20}\).

Understanding Probability in Context

In classrooms, board games, and even weather forecasts, probability helps us make informed predictions and decisions. In our exercise, mastering the concept allows us to approach random selection with a clear sense of chance, clarifying just how likely it is to pick an odd prime number from a shuffled stack of 20 numbered cards.