Problem 17 - Algebra 2

Question

A die is rolled twice. Find each probability. \(P(\text { two different numbers) }\)

Step-by-Step Solution

Answer

The probability of rolling two different numbers is \( \frac{5}{6} \).

Step 1: Understand the Probability Problem

We're tasked with finding the probability of rolling two different numbers on a six-sided die. We'll first outline the total possible outcomes when a die is rolled twice.

Step 2: Determine Total Possible Outcomes

When a die is rolled twice, each roll has 6 possible outcomes. Therefore, the total number of possible combinations is given by multiplying the number of outcomes of each roll: \[6 \times 6 = 36\] outcomes.

Step 3: Identify Favorable Outcomes

To have two different numbers, the numbers on both rolls should differ. Consider the following for each first roll outcome: it can match 1 of the six numbers. To ensure difference, 5 other numbers can be paired with it. Thus, for each of the 6 initial rolls, the second roll has 5 favorable outcomes:\[6 \times 5 = 30\] favorable outcomes.

Step 4: Calculate Probability

Probability is the ratio of favorable outcomes to total possible outcomes. Thus, the probability \(P\) of rolling two different numbers is:\[P(\text{two different numbers}) = \frac{30}{36}\] Simplify the fraction:\[= \frac{5}{6}\].

Key Concepts

Outcomes

When working with probabilities, the term "outcomes" refers to every possible result that may occur during a probability experiment. In the context of the example provided, we have a six-sided die being rolled twice. Each roll of the die offers 6 different possibilities, resulting in a combination of results or outcomes for each scenario.

To calculate the total outcomes when a die is rolled twice, you multiply the number of outcomes of the first roll by the number of outcomes of the second roll. This is because each result of the first roll can be paired with any of the results from the second roll.

• First Roll: 6 possibilities (1 through 6)
• Second Roll: 6 possibilities (1 through 6)
• Total Outcomes: \(6 \times 6 = 36\)

Thus, there are 36 possible outcomes when a die is rolled twice, and this forms the foundation for calculating further probabilities.

Favorable Outcomes

While outcomes encompass all possible results of an event, favorable outcomes are those specific results that match the criteria of the event in question. In the case of rolling two different numbers, favorable outcomes are those where the numbers on both dice differ.

For each result of the first roll, to ensure two different numbers, there are five numbers on the second roll that can pair differently. This is because if the first roll is a 1, the second roll must be 2, 3, 4, 5, or 6. Continuing this pattern:

• First Roll: 6 possibilities (1 through 6)
• Favorable Outcomes per First Roll: 5 different outcomes
• Total Favorable Outcomes: \(6 \times 5 = 30\)

Therefore, out of the 36 total outcomes, 30 outcomes are those where two different numbers are rolled.

Simplifying Fractions

In probability, expressing the result as a simplified fraction can help in understanding the likelihood of the event. After determining the total number of outcomes and favorable outcomes, we calculate the probability using the ratio of these two figures.

Using our previous calculations, we found that there are 30 favorable outcomes out of 36 total outcomes. The probability expression is \(\frac{30}{36}\).

To simplify the fraction, find the greatest common divisor of the numerator and the denominator and divide both by it:

• Numerator: 30
• Denominator: 36
• Greatest Common Divisor: 6 (since both can be divided evenly by 6)
• Simple Form: \(\frac{30 \div 6}{36 \div 6} = \frac{5}{6}\)

This means the probability of rolling two different numbers is \(\frac{5}{6}\), giving a simple and clear representation of how often this event happens compared to all possible events.