Problem 16 - Algebra 2

Question

A die is rolled twice. Find each probability. \(P(\text { two of the same number) }\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{6}\).

Step 1: Understand the Problem

We are asked to find the probability of rolling the same number on both rolls when a die is rolled twice. This means the outcome of the first roll must match the outcome of the second roll.

Step 2: Determine Total Outcomes

A die has 6 sides, so there are 6 possible outcomes for each roll. When rolling a die twice, the total number of possible outcomes is the product of the outcomes for each roll, calculated as \(6 \times 6 = 36\).

Step 3: Identify Favorable Outcomes

Two dice show the same number if both come up as 1, or both as 2, and so on until both show 6. Therefore, the favorable outcomes are \((1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\), making a total of 6 favorable outcomes.

Step 4: Calculate Probability

The probability of rolling the same number on both dice is the number of favorable outcomes divided by the total number of possible outcomes. Hence, the probability \(P\) is calculated as \(P = \frac{6}{36} = \frac{1}{6}\).

Key Concepts

Sample Space

When engaging in probability problems, understanding the concept of a sample space is fundamental. A **sample space** is the set of all possible outcomes of a probabilistic experiment. In this case, the experiment consists of rolling a die twice. Each roll results in one of six possible outcomes, ranging from 1 to 6.

For multiple independent events, such as rolling a die twice, the sample space is the Cartesian product of the individual sample spaces for each event. Thus, the total sample space for rolling a die twice is represented by all possible pairs: (1,1), (1,2), ..., (6,6). There are 36 such pairs, as each first roll can be paired with each outcome of the second roll. Recognizing the sample space helps us establish a foundation to further determine specific outcomes and probabilities.

Favorable Outcomes

In any probability question, identifying the **favorable outcomes** is crucial. These are the outcomes that fulfill the conditions of the event we are focusing on.

For the problem at hand, we are interested in the scenario where both dice show the same number. Thus, favorable outcomes are pairs where the numbers are identical, such as (1,1), (2,2), ..., (6,6).

• (1,1)
• (2,2)
• (3,3)
• (4,4)
• (5,5)
• (6,6)

These six outcomes directly relate to the condition we want to satisfy, and understanding them is key to calculating the probability of this event happening.

Total Outcomes

Calculating probability requires knowing both the favorable outcomes and the **total outcomes**. The total outcomes give the complete picture of possibilities for an event to happen.

In our case, when rolling a die twice, each roll has 6 possible outcomes. Since these events are independent (the result of the first roll doesn't affect the second), the total number of outcomes is the product of the possibilities for each roll. Hence, we have:

• First roll: 6 possible outcomes
• Second roll: 6 possible outcomes

That's a total of 6 x 6, resulting in 36 possible outcomes.

This number serves as the denominator in our probability fraction. By dividing the number of favorable outcomes by this total, we determine the probability of the event occurring.