Problem 49 - Algebra and Trigonometry

Question

A die is rolled twice. What is the probability of showing a one on the first roll and an even number on the second roll?

Step-by-Step Solution

Answer

The probability is \( \frac{1}{12} \).

Step 1: Identify Outcomes for the First Roll

When rolling a die, there are six possible outcomes: 1, 2, 3, 4, 5, and 6. We are interested in getting a 1 on the first roll. The probability of rolling a 1 on a fair six-sided die is \( \frac{1}{6} \).

Step 2: Identify Outcomes for the Second Roll

For the second roll, we need an even number. The even numbers on a die are 2, 4, and 6. This gives us three favorable outcomes out of six possible. Thus, the probability of rolling an even number is \( \frac{3}{6} = \frac{1}{2} \).

Step 3: Calculate the Combined Probability

The events of rolling a die twice are independent. Hence, the probability of rolling a 1 on the first roll and then an even number on the second roll is the product of the two individual probabilities: \( P(1 \,\text{on first and even on second}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \).

Key Concepts

Independent Events

In probability theory, independent events are events where the outcome of one event does not affect the outcome of another. Understanding independent events is crucial for determining probabilities in situations involving multiple actions, like rolling dice multiple times.
For instance, when we roll a die twice, the result of the first roll does not influence the result of the second roll. Each roll has its full set of possibilities.

• When you roll a die, you have six potential outcomes.
• Rolling a 1 in the first trial does not alter the possibilities for the second trial.
• Therefore, each roll remains an independent event.

Recognizing these independent events allows us to use specific rules in probability like the multiplication rule.

Probability of Dice Rolls

Dice rolls are a classic example used in probability theory due to their simple structure and fixed number of outcomes. A standard die has six faces, each equally likely if the die is fair. Thus, the probability of any single face showing up on a roll is consistent and straightforward to calculate.
For any specific number appearing on a roll of a fair six-sided die:

1. There is always a probability of \( \frac{1}{6} \), since there is one favorable outcome over six possible outcomes.
2. For rolling even numbers, like 2, 4, or 6, the probability changes because there are three favorable numbers.
3. The probability becomes \( \frac{3}{6} = \frac{1}{2} \), indicating a broader range of favorable outcomes.

Dice provide an excellent foundation for exploring probability because of their fixed and clear-cut outcomes.

Multiplication Rule for Probability

The multiplication rule for probability is a powerful tool used when determining the probability of two or more independent events occurring in sequence. This rule allows you to compute the joint probability by multiplying the probabilities of each independent event.
For example, if you want to find the probability of rolling a 1 on the first die and an even number on the second, the events are independent, and the multiplication rule applies.

• The probability of rolling a 1 is \( \frac{1}{6} \).
• The probability of rolling an even number is \( \frac{1}{2} \).

To find the combined probability, you multiply these individual probabilities: \[ P(1 \text{ on first and even on second}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}. \] This resulting probability reflects the combined effect of both actions occurring together, showcasing the utility and simplicity of the multiplication rule in probability calculations.