Problem 14 - College Algebra

Question

A die is rolled. Find the probability of getting a number greater than 3

Step-by-Step Solution

Answer

The probability of rolling a number greater than 3 is 0.5.

Step 1: Count the Total Possible Outcomes

For a standard six-sided die, there are 6 possible outcomes: {1, 2, 3, 4, 5, 6}.

Step 2: Identify the Favorable Outcomes

The problem asks for the probability of rolling a number greater than 3. The numbers greater than 3 are {4, 5, 6}. Therefore, there are 3 favorable outcomes.

Step 3: Calculate the Probability

Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. For this problem, the probability (P) can be calculated using the formula \( P = \frac{{\text{{Number of favorable outcomes}}}}{{\text{{Total number of outcomes}}}} \). Substituting the given values, the formula becomes \( P = \frac{3}{6} = 0.5 \).

Key Concepts

Die Roll Probability

Die roll probability is simply the chance of a particular outcome occurring when a die is rolled. In the context of a standard six-sided die, these probabilities are based on the fact that each of the six faces is equally likely to land face-up. This is because an ideal die is symmetric, meaning each side has the same chance of appearing after a roll.

When we express probability, it's often in the form of a fraction, decimal, or percentage. For standard die rolls, the probability will always have a denominator of 6 because there are 6 possible outcomes. For example, the probability of rolling exactly a 3 would be represented as \( \frac{1}{6} \) or approximately 0.167, which is about 16.7%.

Favorable Outcomes

In probability, 'favorable outcomes' are the specific results of an event that we are interested in. For example, if we want to roll a die and hope to get a number greater than 3, the favorable outcomes would be rolling a 4, 5, or 6. These outcomes are 'favorable' because they meet the criteria we have set.

Identifying favorable outcomes is a critical step since it directly influences the probability calculation. In our die-rolling example, there are 3 favorable outcomes. It is important to count only the unique results that satisfy the desired condition when determining the count of favorable outcomes.

Total Possible Outcomes

The 'total possible outcomes' refer to all the different results that could possibly occur when an event, such as rolling a die, takes place. For a six-sided die, there are six total possible outcomes because the die can land on any one of the six faces.

The concept of 'total possible outcomes' establishes the fundamental sample space or the 'universe' of all possible occurrences for a particular random event. When calculating probabilities, this is the denominator in the probability fraction, as it represents the complete set of equally probable outcomes. It is crucial to have a thorough understanding of the total possible outcomes to ensure accurate probability calculations.