Problem 15 - Algebra 2

Question

A die is rolled twice. Find each probability. \(P(1, \text { then any number) }\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Event

To solve for the probability of the event where a die shows a 1 on the first roll and any number on the second roll, identify the sample space for each roll of the die. A single die has 6 sides and may show any number from 1 to 6.

Step 2: Determine the Events for Each Roll

The first roll needs to show a 1, which is only 1 possible outcome ( 1 ). The second roll can show any number from 1 to 6, so there are 6 possible outcomes for the second roll.

Step 3: Calculate the Probability for Each Roll

The probability of rolling a 1 on the first roll is: \[ P( ext{roll} = 1) = \frac{1}{6} \]The probability of rolling any number on the second roll is: \[ P( ext{any number}) = 1 \] since every roll of the die results in any number from 1 to 6.

Step 4: Calculate the Combined Probability

Since these are independent events, the combined probability is the product of both probabilities:\[ P(1, \text{ then any number}) = \frac{1}{6} \times 1 = \frac{1}{6} \]

Key Concepts

Independent Events

When we talk about independent events in probability, we refer to events where the outcome of one event does not affect the outcome of another. For example, rolling a die two times involves independent events because the result of the first roll has no impact on the result of the second roll.

In technical terms, two events, A and B, are considered independent if the probability of both events occurring is the product of the probabilities of each event happening separately. Mathematically, this can be expressed as:

• \(P(A \text{ and } B) = P(A) \times P(B) \)

Understanding independence helps simplify the calculation of probabilities, especially in situations involving multiple trials or actions. Thus, for dice, card games, and many real-life applications, knowing this concept is crucial for solving probability problems.

Sample Space

Sample space is a fundamental concept in probability that refers to the set of all possible outcomes of a random experiment. When you roll a single die, the sample space is all the numbers that the die can show, which are 1, 2, 3, 4, 5, and 6.

If you're doing more complex experiments, say rolling a die twice, the sample space becomes all possible combinations of results from both rolls. For each roll of the die, the sample space remains the same with six possible outcomes.

To illustrate further:

• First roll sample space: \(\{1, 2, 3, 4, 5, 6\}\)
• Second roll sample space: \(\{1, 2, 3, 4, 5, 6\}\)

Keeping track of the sample space is crucial because it allows us to calculate probabilities accurately and is the foundation for understanding how different outcomes can occur in probability.

Rolling Dice

Rolling dice is a common activity used to teach and understand basic probability. Each roll of a standard die has an equal likelihood of landing on any number from 1 to 6. This is what makes rolling dice a fair random event.

The probability of any single outcome, such as rolling a 1, is given by dividing the number of favorable outcomes by the total number of possible outcomes. For a single die:

• Probability of rolling a specific number (e.g., 1): \( P(\text{specific number}) = \frac{1}{6} \)

Understanding how dice rolls work in probability helps illustrate the concept of equal likelihood and paves the way for more complex scenarios like rolling two dice and calculating the probabilities of combined outcomes.

Combined Probability Calculation

Calculating combined probabilities involves determining the likelihood of two or more events happening together. If the events are independent, as with rolling a die twice, their combined probability is found by multiplying their individual probabilities.

Using the example of rolling a die twice:

• The probability of rolling a 1 on the first roll is \(\frac{1}{6}\)
• The probability of rolling any number on the second roll is 1, since the second roll can result in any number from the sample space

To find the combined probability, multiply the probabilities:

• \(P(1, \text{ then any number}) = \frac{1}{6} \times 1 = \frac{1}{6}\)

By understanding how to combine probabilities, we can solve more complex problems and gain deeper insights into how various events are interrelated within the realm of probability.