Problem 48 - Algebra 2

Question

A die is rolled three times. Find each probability. \(P(1, \text { then } 2, \text { then } 3)\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{216} \).

Step 1: Understanding the Problem

We need to calculate the probability of rolling a 1, then a 2, and then a 3, in that exact order with a fair die.

Step 2: Calculate Probability of Each Roll

The probability of rolling a specific number on a fair six-sided die is \( \frac{1}{6} \). Since the die is rolled three times and each roll is an independent event, we first calculate the probability for each roll independently. So the probability of rolling a 1 on the first roll is \( \frac{1}{6} \), rolling a 2 on the second roll is \( \frac{1}{6} \), and rolling a 3 on the third roll is \( \frac{1}{6} \).

Step 3: Use the Multiplication Rule

Since the rolls are independent, we find the probability of successive events (rolling 1, then 2, then 3) by multiplying the probabilities of each event. The combined probability is \( \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} \).

Step 4: Calculate the Final Probability

Now, perform the multiplication: \( \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} = \frac{1}{216} \). Thus, the probability of rolling a 1, then a 2, and then a 3 with a fair die is \( \frac{1}{216} \).

Key Concepts

Independent Events

In probability, independent events are those where the outcome of one event does not affect the outcome of another. These events are crucial in many scenarios, especially when dealing with sequences of events.

For example, when you roll a die, the number you get on your first roll does not influence the number you will get on your second roll. Each roll of the die is independent of all others. This is true provided no biases are introduced, such as a rigged die or any external interference.

Knowing whether events are independent is important because it determines how we calculate the probability of multiple events happening in sequence. Only independent events allow us to use certain rules, like the multiplication rule, to combine their probabilities.

Multiplication Rule

The multiplication rule is a principle in probability theory that allows us to find the probability of two or more independent events happening in sequence.

To apply the multiplication rule, you multiply the probability of each event occurring. For instance, let's consider rolling a die three times, aiming for different numbers each time.

• Probability of rolling a 1: \( \frac{1}{6} \).
• Probability of rolling a 2, independent of the first roll: \( \frac{1}{6} \).
• Probability of rolling a 3, also independent: \( \frac{1}{6} \).

The multiplication rule tells us that the overall probability is the product of these probabilities: \( \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} = \frac{1}{216} \).

It’s important that the rule applies only when events do not influence each other, reinforcing the concept of independent events.

Rolling a Die

Rolling a die is a simple but effective way to explore basic probability concepts. A standard die has six faces, numbered from 1 to 6.

Each face is equally likely to land face up when you roll the die, assuming the die is fair and there is no deliberate force influencing the outcome. This results in a probability of \( \frac{1}{6} \) for each number.

Dice are often used in probability exercises because they have a clear and finite set of outcomes. This makes them ideal for teaching concepts like independent events and the multiplication rule. When rolling the die several times or in combination with other dice, the outcomes of these rolls can be calculated using these basic principles.

• The probability of rolling any specific number is \( \frac{1}{6} \).
• Each roll is an independent event.
• The dice provide a clear demonstration of probability operations.

Understanding the mechanics of rolling a die is foundational in grasping broader statistical concepts.