Question

In the roll of an ideal die, the probability of getting a prime number is

Short answer

The probability of getting a prime number is \( \frac{1}{2} \).

Step-by-step solution

Understanding Prime Numbers on a Die

An ideal die is a six-sided cube with faces numbered from 1 to 6. We need to identify which of these numbers are prime. A prime number is a number greater than 1 that has no divisors other than 1 and itself. For a six-sided die, the potential numbers are: 2, 3, 4, 5, and 6. Among these, 2, 3, and 5 are prime numbers.

Counting Prime Numbers

From the numbers 1 to 6 on the die, we have identified three prime numbers: 2, 3, and 5. There are three prime numbers in total.

Calculating the Probability

The probability of an event is given by the number of successful outcomes divided by the total number of possible outcomes. Here, the number of successful outcomes (rolling a prime number) is 3, and the total number of possible outcomes is 6 (since there are 6 faces on the die). The probability can be calculated as follows: \( P(\text{prime}) = \frac{3}{6} \).

Simplifying the Probability Fraction

Simplify the fraction \( \frac{3}{6} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This simplifies the fraction to \( \frac{1}{2} \).

Key concepts

Prime Numbers

When discussing probability theory, especially in the context of dice, understanding prime numbers is crucial. A prime number is any number greater than one that cannot be evenly divided by any other number than one and itself. In simpler terms, it has exactly two distinct positive divisors.
For example:

• 2 is prime because the only divisors are 1 and 2.
• 3 is prime as well, since only 1 and 3 divide it.
• 5 is another prime number with divisors being 1 and 5.

On a standard die that shows numbers 1 through 6, you can test each number to see if it's prime. Numbers like 2, 3, and 5 meet the criteria. Learning which numbers are prime helps you predict outcomes, especially in games of chance.

Die Roll

Rolling a die is one of the simplest forms of a random experiment in probability theory. A standard die is a cube with 6 faces, each face showing a different number from 1 to 6. Each roll can result in any of these numbers, making the probability of landing on any one number equal.
In probability terms, each side of the die has a \(rac{1}{6}\) chance of facing up after a roll, assuming the die is fair. The concept of a fair die is important—it implies that each outcome is equally likely. This foundation helps in calculating more complex probabilities, like rolling a prime number.

Mathematical Probability

Mathematical probability helps us determine the likelihood of various outcomes. It's calculated by taking the number of favorable outcomes and dividing it by the total number of possible outcomes. When you're trying to find the probability of rolling a prime number on a die:
- Count the prime numbers that can appear (2, 3, and 5). This gives us 3 favorable outcomes.- The total number of outcomes possible is 6, as there are six faces.
The probability is then expressed as \(rac{3}{6}\) because there are 3 favorable outcomes out of 6 possible. Understanding this basic formula allows you to calculate the probability of other events as well.

Simplifying Fractions

Simplifying fractions is a useful skill in mathematics and probability calculations. It involves reducing a fraction to its simplest form. This is done by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it.
For instance, when calculating probabilities such as \(rac{3}{6}\), you simplify by identifying the GCD, which is 3 in this case. Dividing both the numerator and the denominator by 3 results in \(rac{1}{2}\).
The simplified fraction \(rac{1}{2}\) makes interpretation easier—it shows that there is a one in two chance, or a 50% probability, of rolling a prime number with a standard die. Simplifying results allows conclusions to be clearer and more precise.