Problem 16 - College Algebra

Question

A die is rolled. Find the probability of getting a number greater than 7.

Step-by-Step Solution

Answer

The probability of rolling a number greater than 7 on a regular die is 0.

Step 1: Understand the probability theory

The probability of an event is defined as the number of favourable outcomes divided by the total number of outcomes. A standard die is a cube with 6 faces that are numbered from 1 to 6. Each roll of the die is a separate event and the outcome can be any number from 1 to 6.

Step 2: Identify the favourable outcomes

The event required is to get a number greater than 7. But on a regular die, the highest value that can be rolled is 6. Therefore, there is no favourable outcome in this scenario.

Step 3: Calculate the probability

As there are no favourable outcomes, the probability of rolling a number greater than 7 is 0. In other terms, Probability (P) = \( \frac{Number of favourable outcomes}{Total number of outcomes} \) = \( \frac{0}{6} \) = 0

Key Concepts

Probability Theory

Probability theory is a foundational element in statistics and mathematics. It gives us a systematic way to determine how likely events are to happen. When we talk about probability, we often express it as a fraction or percentage that ranges between 0 and 1, where 0 means an event will not happen, and 1 means it will definitely happen.

To calculate probability, we use the formula \( P = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}} \). This means we look at how many outcomes meet our criteria versus the total number of possibilities.

• *Total number of outcomes* refers to all the possible outcomes that can occur.
• *Favourable outcomes* are the outcomes that align with the event we are interested in.

In many practical problems, identifying these outcomes is the first step in solving probability-related questions.

Favourable Outcomes

Favourable outcomes are crucial when calculating probabilities. They are the specific outcomes of an event that fulfill the condition we are interested in. For examples like rolling a dice, it's important to recognize which numbers represent the outcomes you are focusing on.

In the example from the exercise, the condition was rolling a number greater than 7 with a standard die. As a standard die has numbers from 1 to 6, there are no outcomes that meet the required condition. This means:

• There are 0 favourable outcomes that satisfy the condition (number greater than 7).

If there are no favourable outcomes, the probability of the event is zero, as an event with no favourable circumstances cannot occur.

Dice Probability

Dice probability is a common topic when discussing probability due to the simplicity and familiarity of dice games. A standard die has six faces, each marked with a distinct number from 1 to 6. This makes calculating probabilities with a die straightforward.

For any single roll of the die, there are 6 possible outcomes. Each face is equally likely to show up, assuming the die is fair. When determining the probability of events with a die, you follow these steps:

• Define the event: What are you looking to observe?
• Identify the favourable outcomes among the 6 faces.
• Use the probability formula \( P = \frac{\text{Number of favourable outcomes}}{6} \).

For instance, the probability of rolling a 3 is \( \frac{1}{6} \), because there is one 3 on a six-faced die. Yet, if the event was to roll a number greater than 7, it would be \( \frac{0}{6} = 0 \), since such an outcome is impossible.