Question

The probability of getting 2 or 3 or 4 from a throw of single dice is...

Short answer

The probability of getting 2, 3, or 4 is \( \frac{1}{2} \).

Step-by-step solution

Identify Possible Outcomes

When throwing a single die, there are 6 possible outcomes: 1, 2, 3, 4, 5, and 6.

Determine Favorable Outcomes

From the possible outcomes, the numbers 2, 3, and 4 are the favorable outcomes for our event.

Calculate Number of Favorable Outcomes

There are three favorable outcomes: 2, 3, and 4.

Calculate Total Number of Possible Outcomes

The total number of possible outcomes when rolling a die is 6.

Calculate Probability

The probability is the ratio of favorable outcomes to the total number of possible outcomes. Thus, the probability of getting a 2, 3, or 4 is given by \( \frac{3}{6} \).

Simplify Probability

Simplify the fraction \( \frac{3}{6} \) to \( \frac{1}{2} \).

Key concepts

Dice Probability

When you roll a single die, you're dealing with an easy-to-understand type of chance, called dice probability. A standard die has six sides, and each side has a different number from 1 to 6. This means there are always six possible outcomes every time you roll it.

In a specific exercise, if we want to find out how likely it is to roll a 2, 3, or 4, we start by considering these three faces of the die as our "favorable outcomes." These are the outcomes we are interested in for this exercise.

For this scenario, each number from 1 to 6 on the die is equally likely to appear. This means that each outcome has a probability of occurring that is equal to the others, thanks to the fairness of the dice roll process.

To find the probability, we simply compare how often our favorable outcomes (2, 3, and 4) can occur to the total number of possible outcomes (which is 6, the six numbers on the die). This helps us measure the likelihood of rolling one of the favorable outcomes.

By dividing the number of favorable outcomes by the total outcomes, we calculate the probability and simplify if needed.

Probability Theory

Probability theory is a branch of mathematics that helps us understand randomness and chance. It's about measuring how likely something is to happen. In the context of rolling a die, probability theory gives us a structured way to assess different outcomes, so we can know just how likely rolling a certain number is.

In essence, probability theory involves some key terms:

• **Experiment**: This is the action we are interested in, such as rolling a die.
• **Outcome**: This refers to the result of the experiment, such as rolling a 2.
• **Favorable Outcome**: These are the specific outcomes we are looking for, like rolling a 2, 3, or 4 in our exercise.
• **Sample Space**: The complete set of all possible outcomes for the experiment, which would be the numbers 1 to 6 when rolling a standard die.

The probability of an event happening is calculated by dividing the number of favorable outcomes by the number of possible outcomes in the sample space. This gives us a number between 0 and 1, where 0 means the event is impossible, and 1 means it is certain. For example, the probability of rolling a 2, 3, or 4 with a single die is simplified from \( \frac{3}{6} \) to \( \frac{1}{2} \). This indicates a 50% chance of the event occurring.

Favorable Outcomes

Favorable outcomes are specific results from an event that we are concerned with or interested in. When calculating probability, identifying these outcomes is crucial. In a dice-related problem, favorable outcomes directly affect the probability calculation.

Suppose you're rolling a die and you want to know the chance of getting a number that is 2, 3, or 4. The numbers you want, which are the 2, 3, and 4, are your favorable outcomes.

Here's how identifying favorable outcomes works in practice:

• First, list out all possible outcomes, which for a six-sided die will always be 1, 2, 3, 4, 5, and 6.
• Next, pinpoint which of these outcomes are favorable to your event. In this case, there are three: 2, 3, and 4.
• Counting these favorable numbers gives you a sense of their likelihood. Since there are three favorable outcomes out of a total of six possible outcomes, the calculation is \( \frac{3}{6} \).

The concept of favorable outcomes is straightforward but critical, as it drives how we calculate and understand the probability in any given scenario. Identifying these outcomes allows you to zero in on what exactly you care about in a probability exercise.