Problem 59 - Algebra 2

Question

A standard number cube is tossed. Find each probability. \(P(\text { prime or } 2)\)

Step-by-Step Solution

Answer

The probability of getting a prime number or a number equal to 2 when a standard number cube is tossed is \(\frac{1}{2}\).

Step 1: Identify Possible Outcomes

The possible outcomes of tossing a standard cube are 1, 2, 3, 4, 5, 6. Here, prime numbers are 2, 3 and 5 only. Hence outcomes for prime numbers or equal to 2 are 2, 3 and 5.

Step 2: Calculate the Probability

Probability is the ratio of the number of successful outcomes to the total number of outcomes. The total number of outcomes when a cube is tossed is 6. The successful outcomes are 2, 3 and 5. So the number of successful outcomes is 3. Therefore, the probability is calculated as the number of successful outcomes divided by the total number of outcomes. So, \(P(\text {prime or } 2) = \frac{3}{6}\)

Step 3: Simplify the Probability

The fraction \(\frac{3}{6}\) simplifies to \(\frac{1}{2}\). Therefore, the probability of rolling a prime number or a number equal to 2 is \(\frac{1}{2}\)

Key Concepts

Number Cube

A number cube is just another term for a die used in various games. It is a cube with six faces, each displaying a different number from 1 to 6.

• When you roll a number cube, each number from 1 to 6 has an equal chance of landing on top.
• This makes the number cube a perfect example for studying probability, as each face is equally likely to appear.

When solving probability problems with a number cube, it's important to list all possible outcomes, which in this case are the numbers 1 through 6. This forms the basis of calculating probabilities.

Prime Numbers

Prime numbers are those special numbers greater than 1 that have no divisors other than 1 and themselves.

• In the context of a number cube, the numbers 2, 3, and 5 are prime.
• These numbers cannot be divided evenly by anything other than 1 and the number itself.

Understanding which numbers are prime helps in identifying successful outcomes in probability problems, especially when the problem involves dice or number cubes. In this exercise, when a number cube is rolled, finding prime numbers helps narrow down the favorable outcomes.

Successful Outcomes

The term successful outcomes refers to the specific results that fulfill our probability criteria. For example, if we are looking for prime numbers or a 2, the successful outcomes on a number cube are the numbers 2, 3, and 5.

• First, identify what outcomes are considered 'successful' based on the problem.
• Make sure to count them accurately to determine how many successful outcomes are possible.

In this scenario, there are three successful outcomes: rolling a 2, 3, or 5. Knowing this helps you set up the probability problem correctly.

Fraction Simplification

Fraction simplification is a vital skill in probability and mathematics broadly. It involves reducing a fraction to its simplest form.

• To simplify a fraction, divide both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).
• In our problem, the probability starts as a fraction \(\frac{3}{6}\).

The GCD of 3 and 6 is 3. Dividing both by 3 simplifies the fraction to \(\frac{1}{2}\). This process makes interpreting probability results simpler and more straightforward.