Problem 12 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The probability is \(\frac{1}{36}\).

Step 1: Understand the Event

We need to find the probability of rolling a 2 on the first roll and then rolling a 3 on the second roll of a die. Each roll is independent, meaning the outcome of one does not affect the other.

Step 2: Calculate the Probability of Rolling a 2 First

Consider the first roll of the die. There are 6 possible outcomes (1 through 6). The probability of rolling a 2 is therefore \( \frac{1}{6} \).

Step 3: Calculate the Probability of Rolling a 3 Second

Consider the second roll of the die. Again, there are 6 possible outcomes. The probability of rolling a 3 is also \( \frac{1}{6} \).

Step 4: Use the Multiplication Rule for Independent Events

Since the rolls are independent, the probability of both events happening (rolling a 2 first and a 3 second) is the product of their probabilities: \( \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \).

Key Concepts

Independent Events

In the world of probability, understanding independent events is crucial. When we say that two events are independent, it simply means the outcome of one event does not influence the outcome of the other. Think of it as tossing a coin and rolling a die – no matter how the coin lands, it won't affect the number you roll.
For example, if you roll a die twice, the result of the first roll doesn't affect the result of the second roll. Each roll starts fresh. This is why the probability of rolling a specific number remains the same each time you roll.
Real-life examples of independent events include:

• Flipping a coin and drawing a card from a deck
• Rolling a die and spinning a spinner
• The weather tomorrow and rolling a die today

Recognizing independence helps us to correctly calculate probabilities, especially when dealing with multiple events.

Multiplication Rule

When faced with independent events, the multiplication rule is a powerful tool for calculating probabilities. This rule states that for two independent events, the probability of both events happening is the product of their individual probabilities.
For instance, if you want to find the probability of rolling a 2 and then a 3 on a die, you multiply the probability of rolling a 2 (\( \frac{1}{6} \)) by the probability of rolling a 3 (\( \frac{1}{6} \)). The result is \( \frac{1}{36} \).
Here are steps to use the multiplication rule:

• Determine if events are independent
• Find individual probabilities for each event
• Multiply the probabilities

The multiplication rule is straightforward and helps simplify problems involving multiple events.

Dice Probability

Dice probability is a common topic when exploring the fundamentals of probability. A standard die has six faces, each showing a different number from 1 to 6. Each face has an equal chance of landing face up when rolled.
Calculating the probability of a specific event with a die is simple because the number of favorable outcomes is divided by the total number of outcomes. For example, the probability of rolling a 2 is: \( \frac{1}{6} \).
With two dice rolls, you can find probabilities by considering the total number of possible outcomes. Since each roll has 6 outcomes and the rolls are independent, a sequence of two rolls results in 36 (i.e., 6 times 6) possible combinations of outcomes.
Key points to remember about dice probability include:

• Each face has an equal chance of \( \frac{1}{6} \)
• Multiple rolls are independent events
• Combination probabilities can be calculated using multiplication

Dice probability exercises enhance understanding of both simple and complex probability concepts.