Question

In a single throw of a single die, find the probability of obtaining either a 2 or a 5

Short answer

The probability of obtaining either a 2 or a 5 in a single throw of a single die is \(\frac{1}{3}\), which is approximately 33.33%.

Step-by-step solution

Determine the total number of possible outcomes

A single die has six faces, each with a number from 1 to 6. When we throw the die, we can obtain any of these six numbers. Therefore, the total number of possible outcomes is 6.

Determine the number of successful outcomes

In this case, the successful outcomes correspond to obtaining either a 2 or a 5. There are only two such outcomes, so the number of successful outcomes is 2.

Calculate the probability

The probability of an event is calculated using the following formula: \(P(\text{event}) = \frac{\text{number of successful outcomes}}{\text{total number of possible outcomes}}\) Using the values we determined in steps 1 and 2, we can calculate the probability of obtaining either a 2 or a 5: \(P(\text{2 or 5}) = \frac{2}{6}\)

Simplify the probability

Now, we can simplify the fraction \(\frac{2}{6}\) by dividing the numerator and the denominator by their greatest common divisor, which is 2: \(P(\text{2 or 5}) = \frac{2}{6} = \frac{1}{3}\)

Interpret the result

The probability of obtaining either a 2 or a 5 in a single throw of a single die is \(\frac{1}{3}\). This means that there is a 1 in 3 chance of this event occurring, or approximately 33.33% chance.

Key concepts

Successful Outcomes

In probability, "successful outcomes" refer to the outcomes that satisfy the condition of a particular event. They are the specific results we are interested in. In our exercise, our focus is on the event of throwing a die and obtaining either a 2 or a 5. When we roll a die, there are six possible numbers that can be the result, from 1 to 6.

Successful outcomes in this scenario are limited to the numbers 2 and 5. Thus, the number of successful outcomes in our case is exactly two since only a 2 or a 5 meets the criteria of the event we are examining. Understanding and identifying successful outcomes is the first step towards calculating the probability of any event.

Simplifying Fractions

To simplify a fraction means to make it as simple as possible, achieving a reduced form where the numerator and the denominator have no common divisors other than 1. Simplifying fractions is an essential mathematical process that helps to make calculations and interpretations easier.When finding probability, it is common to end up with fractions that can often be simplified. In our example, the initial probability of rolling a 2 or a 5 is represented by the fraction \( \frac{2}{6} \).

To simplify, identify the greatest common divisor (GCD) of the numerator and denominator. Here, both 2 and 6 can be divided by 2, making the GCD 2. We divide both the numerator and the denominator by this GCD:\[\frac{2 \div 2}{6 \div 2} = \frac{1}{3}.\]Simplifying fractions helps us to understand probability better, offering a clearer picture of the likelihood of outcomes.

Total Outcomes

The concept of "Total Outcomes" relates to the number of possible results that can occur in a given scenario. When calculating probability, it's vital to identify the total number of outcomes as it forms the denominator in the probability formula. In the context of a single die throw, the "total outcomes" are all the possible numbers that can appear on the upper face of the die. A standard die has six faces, numbered from 1 to 6. Hence, there are a total of six possible outcomes when rolling a single die.

Recognizing the total number of potential results is crucial because it serves as the baseline for determining the proportion of successful outcomes, thereby forming the foundation of any probability problem.

Dice Probability

Dice probability involves analyzing and calculating the likelihood of different outcomes when rolling one or more dice. Dice are commonly used in games, making understanding their probability crucial for both strategic play and academic contexts.A single, six-sided die generates six equiprobable outcomes, meaning each face—numbered 1 through 6—has an equal chance of landing face up. The probability of any specific outcome can be calculated by dividing the number of that specific outcome by the total number of outcomes (6 for a single die).

In the given exercise, we calculate the probability of rolling a 2 or a 5 with a single die. Expressing this as a formula:\[P(\text{2 or 5}) = \frac{\text{Number of successful outcomes}}{\text{Total outcomes}} = \frac{2}{6} = \frac{1}{3}\]Thus, dice probability helps us understand the fairness, odds, and strategic aspects of games involving dice, while also being a fundamental probability topic in mathematics.