Problem 59 - 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: Understand the problem

The problem asks us to find the probability of rolling a one on the first roll of a die and an even number on the second roll. A standard die has six faces numbered from 1 to 6.

Step 2: Calculate Probability for the First Roll

For the first roll of the die, we want to know the probability of rolling a 1. There is only one face with a 1 on a die. Therefore, the probability of rolling a 1 is \( \frac{1}{6} \).

Step 3: Calculate Probability for the Second Roll

For the second roll, we need the probability of rolling an even number. The even numbers on a die are 2, 4, and 6. There are three even numbers, so the probability is \( \frac{3}{6} \) or \( \frac{1}{2} \).

Step 4: Calculate Combined Probability

The probability of both events occurring (rolling a 1 first and an even number second) is the product of the individual probabilities. Therefore, the combined probability is \( \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \).

Step 5: Finalize the Answer

By multiplying the probabilities from Steps 2 and 3, we have found the probability of the desired sequence of rolls: \( \frac{1}{12} \).

Key Concepts

Independent events

In probability, independent events are those events where the outcome of one event does not affect the outcome of another event. When we are dealing with independent events, like rolling a die, each roll is an independent event because the result of one roll doesn't influence the next.
For example, if you roll a die twice, the result of the first roll does not impact the result of the second roll. This is what makes the events independent.
It’s important because when events are independent, you can find the probability of both events happening by multiplying their individual probabilities.

Probability of a sequence

Finding the probability of a sequence of independent events involves multiplying the probabilities of each event. This means if you have two events in a sequence, like rolling a 1 on the first roll and then rolling an even number on the second roll, you calculate the probability of each separately and then multiply them.
For example:

• Probability of rolling a 1 is \( \frac{1}{6} \)
• Probability of rolling an even number (2, 4, or 6) is \( \frac{3}{6} = \frac{1}{2} \)

The probability of both events occurring in sequence is \( \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \). This sequence shows how you can measure chances for combinations of independent events.

Dice and probability

Dice are a great example when learning about probability. A standard 6-sided die has faces numbered from 1 to 6. Each face has an equal chance of landing face up, each with a probability of \( \frac{1}{6} \).
When calculating probabilities with dice, consider the number of favorable outcomes over the total possible outcomes.
For instance, when finding the probability of rolling an even number, you count the even numbers (2, 4, 6) which are three in total. Therefore, the probability is \( \frac{3}{6} \) or \( \frac{1}{2} \). This is crucial for calculating probabilities in sequences or combinations.

Multiplication rule of probability

The multiplication rule of probability assists in finding the probability of combined independent events. It states that the probability of both independent events occurring is the product of their individual probabilities. This rule is essential when dealing with problems involving multiple events happening in a sequence.
For instance, when calculating the probability of rolling a 1 first and an even number second with a die, you use the multiplication rule:

• First Event: Probability of 1 is \( \frac{1}{6} \)
• Second Event: Probability of even number is \( \frac{1}{2} \)

Multiply these: \( \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \).
This rule makes solving combined probability questions straightforward and manageable.