Problem 23 - Algebra 2

Question

A standard number cube is tossed. Find each probability. \(P(\text { even or prime) }\)

Step-by-Step Solution

Answer

The probability of rolling a number that is either even or prime is \(\frac{5}{6}\)

Step 1: Identifying even numbers

First, identify the even numbers on the cube, which are 2, 4 and 6.

Step 2: Identifying prime numbers

Next, identify the prime numbers on the cube, which are 2, 3 and 5.

Step 3: Identifying common numbers

Find the numbers that are common in both categories - even and prime. The number 2 is both even and prime.

Step 4: Identifying all unique numbers that are either even or prime

Combine all the numbers from steps 1 and 2, and remove the duplicates. This gives us 2, 3, 4, 5, and 6.

Step 5: Calculate the probability

To find the probability, divide the total amount of successful outcomes (either even or prime) by the total number of outcomes. So we have 5 successful outcomes and 6 total outcomes.

Step 6: Expressing the probability

Express this as a fraction, so the probability \(P(\text { even or prime }) = \frac{5}{6}\)

Key Concepts

Prime Numbers

Prime numbers are fascinating components of mathematics, and understanding them is essential for various calculations, including probabilities. A prime number is one that has no divisors other than 1 and itself. In simpler terms, it can't be evenly divided by any other number. The number cube, often referred to as a standard six-sided die, contains numbers from 1 to 6. From these, the List of prime numbers includes 2, 3, and 5.

A key characteristic of prime numbers is that they are greater than 1. However, not all odd numbers are prime. For instance, while 3 and 5 are prime, the number 6 is not, since it can be divided by 2 and 3.

Understanding prime numbers allows you to evaluate events in probability exercises where identifying numbers with specific properties, such as being divisible only by themselves and one, is required. This foundational understanding leads directly into calculating outcomes like the one in the exercise.

Even Numbers

Even numbers are numbers divisible by 2 without a remainder. They form an integral part of probability and statistical analysis. On a typical number cube, the even numbers present are 2, 4, and 6.

Each time you roll a number cube, you have a 1 in 6 chance of landing on one of these even numbers. Understanding even numbers is crucial when you are tasked with identifying probable outcomes in probability exercises.

It's important to remember that while all even numbers can be divided by 2, not all even numbers are prime. In the context of a number cube, the only number that is both even and prime is 2, making it a unique bridge between both categories.

Number Cube

A number cube, commonly known as a die, is a simple tool but plays a significant role in probability exercises. It consists of six faces, each displaying a different number between 1 and 6.

When considering probability problems, each face of the cube has an equal likelihood of appearing when rolled. This means each outcome has a probability of 1/6.

Understanding the layout of a number cube is vital when calculating probabilities related to certain traits like being 'even' or 'prime.' When dealing with combined probabilities, such as finding the probability of rolling an even or prime number, the number cube's properties are foundational to breaking down the problem. This conceptual grounding will guide you in evaluating combined probabilities, simplifying complex probability questions into more manageable calculations.