Problem 34 - Algebra 2

Question

A die is rolled, Find each probability. \(P(\text { prime number })\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{2}\).

Step 1: List Possible Outcomes of a Die

A standard die has six faces, with numbers 1 through 6 on them. So the possible outcomes when a die is rolled are: 1, 2, 3, 4, 5, and 6.

Step 2: Identify Prime Numbers

Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. From the list of numbers on the die (1 to 6), the prime numbers are 2, 3, and 5.

Step 3: Count Prime Number Outcomes

From the possible outcomes of rolling the die (1, 2, 3, 4, 5, 6), the prime numbers are 2, 3, and 5. So there are 3 prime number outcomes.

Step 4: Calculate Total Number of Outcomes

Since a die has six faces, there are 6 possible outcomes when a die is rolled.

Step 5: Calculate Probability

The probability of rolling a prime number is the number of prime number outcomes divided by the total number of outcomes. So, the probability is \(\frac{3}{6}\).

Step 6: Simplify the Probability Fraction

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

Key Concepts

Prime Numbers

Prime numbers are fascinating! They are numbers greater than 1 that can only be divided by 1 and themselves without leaving a remainder.
In other words, they have exactly two distinct positive divisors: 1 and the number itself.
This unique property makes them essential in various fields, including cryptography and number theory.

• It's important to recognize that the number 1 is not a prime number.
• Examples of prime numbers include 2, 3, 5, 7, 11, and so on.
• Notice that 2 is the only even prime number, as any other even number can be divided by 2.

When considering a standard six-sided die, as in our exercise, the primes that can appear on the die's faces are: 2, 3, and 5. Recognizing these numbers quickly is crucial for calculating probabilities accurately.

Outcomes

In probability, the term "outcomes" refers to the possible results that can occur from an experiment or event.
For instance, when rolling a standard die, you can get any of the numbers from 1 to 6, describing a total of six outcomes.

• Each face of the die represents one unique outcome.
• This means, there's an equal chance of getting any of these numbers.

Understanding outcomes helps you frame the problem of probability correctly. When asked for the probability of rolling a prime number, we first had to account for all possible outcomes, which reinforced the basis for calculating probabilities.

Fractions

Fractions are an essential concept in conveying probabilities. They help us express the part of a whole, which is crucial when dealing with chances and likelihoods.
A probability is often represented by a fraction, where the numerator signifies the specific number of favorable outcomes, and the denominator represents the total number of possible outcomes.
For example, when calculating the probability of rolling a prime number on a die:

• The fraction started as \(\frac{3}{6}\), meaning 3 prime outcomes out of 6 total outcomes.
• It succinctly communicates the proportion of favorable outcomes relative to all possible outcomes.

Grasping fractions' role makes understanding and working with probabilities much more intuitive.

Simplification

Simplification is a straightforward but significant process that involves reducing a fraction to its simplest form.
This is done by dividing both the numerator and the denominator by their greatest common factor (GCF).
In our example:

• The original fraction \(\frac{3}{6}\) was simplified by dividing both 3 (numerator) and 6 (denominator) by 3, which is their GCF.
• After simplification, we obtained \(\frac{1}{2}\), which is a clearer and more concise way to express the same probability.

Simplification makes fractions easier to understand and compare, reinforcing clarity and accuracy in communicating mathematical concepts. Whether in math homework or real-life applications, always simplify.