Problem 47 - College Algebra

Question

A single die is rolled twice. Find the probability of getting: an even number the first time and a number greater than 2 the second time.

Step-by-Step Solution

Answer

The probability of rolling an even number on the first throw and getting a number greater than 2 on the second throw is 0.335.

Step 1: Calculate the probability of rolling an even number

On a single die, there are three even numbers out of six possible outcomes. So the probability of rolling an even number can be calculated as \( \frac{Number of Even numbers}{Total number of outcomes} \) which is \( \frac{3}{6} = 0.5 \)

Step 2: Calculate the probability of rolling a number greater than 2

On a single die, there are four possible outcomes that are greater than two. So the probability can be calculated as \( \frac{Number of outcomes greater than 2}{Total number of outcomes} \) which is \( \frac{4}{6} = 0.67 \)

Step 3: Multiply the probabilities for the final answer

The final probability is the product of the two event probabilities, because we want both events to happen. Hence, the combined probability is \( 0.5 * 0.67 = 0.335 \)

Key Concepts

Rolling a Die

Rolling a die is a common exercise in probability theory. It involves understanding the possible outcomes when a standard six-sided die is thrown. A die has six faces, each marked with a different number from 1 to 6.
This makes the probability of any single face appearing, such as rolling a 3 or a 6, equal if the die is fair.

• Each side of the die has a probability of \( \frac{1}{6} \) of being rolled.
• For example, to roll an even number, you look at the even sides: 2, 4, and 6. So, the probability is \( \frac{3}{6} = 0.5 \).

Understanding this simple probability helps to build the foundation for more complex probability concepts.

Compound Events

In probability, compound events involve the combination of two or more individual events. For the exercise at hand, this means rolling a die twice and considering both outcomes together.
To calculate the probability of compound events, multiply the probability of each separate event. Make sure the events are independent of each other.

• For instance, if the goal is to roll an even number first, the probability is \( 0.5 \) as previously calculated.
• The probability of rolling a number greater than 2 on the second roll is \( 0.67 \).

Hence, the combined or compound event probability is the product of these two probabilities: \( 0.5 \times 0.67 = 0.335 \).
Compound probability plays a significant role in more complex scenarios, such as games with multiple dice or cards.

Mathematics Education

Mathematics education often incorporates probability exercises to help students understand real-world applications of mathematical concepts. Engaging with problems like rolling dice encourages logical thinking and reasoning skills.
Students learn to collect data, assess possibilities, and calculate outcomes, which are valuable skills both in academics and daily life.

• Exercises include real-life scenarios such as games or lotteries, making the learning process more interesting.
• Visual tools, like probability trees, might be used to help students understand relationships between events.

This focus on practical application ensures that concepts are easier to grasp and retain, laying a strong foundation for further study in statistics or any field requiring analytical skills.