Problem 58 - Algebra and Trigonometry

Question

A die is rolled twice. What is the probability of showing a one on both rolls?

Step-by-Step Solution

Answer

The probability of rolling a one on both dice rolls is \( \frac{1}{36} \).

Step 1: Determine Probability of Rolling a One Once

A die has six faces, each with an equal probability of 1/6 of being rolled. Therefore, the probability of rolling a one on a single roll is \( \frac{1}{6} \).

Step 2: Determine Probability of Independent Events

Since each die roll is independent, the probability of rolling some specific outcome on two rolls is the product of the individual probabilities. Therefore, the probability of rolling a one on both rolls is \( \frac{1}{6} \times \frac{1}{6} \).

Step 3: Calculate the Combined Probability

Multiply the probabilities of each event together to find the combined probability. \( \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \). This is the probability of getting a one on both rolls.

Key Concepts

Independent Events

Understanding independent events is crucial when solving probability problems. In probability theory, events are termed 'independent' if the occurrence of one event does not affect the occurrence of another. For example, when you roll a die, the outcome has no influence on the outcome of a subsequent roll.
This concept is significant because it allows us to calculate the combined probability of two independent events by multiplying their individual probabilities together.
In the context of rolling two dice, each roll is independent. If we want to find the probability of getting certain results on both dice, we simply take the probability of each result and multiply them together.

Dice

Dice are small throwable objects with multiple resting positions. They are commonly used in games and probability exercises. Standard dice are six-sided with numbers ranging from 1 to 6.
When rolling a die, each number has an equal chance of appearing, assuming it's fair and balanced. This gives each side a probability of 1/6. Knowing this basic principle is helpful for predicting outcomes and understanding how probability is distributed across the six faces.

• Fair die: A die where each face has an equal chance of landing face up.
• Biased die: A die with unequal probabilities for different faces, often due to imperfections.

Understanding these concepts of a dice helps when you deal with probability problems involving them.

Rolling a Die

Rolling a die is a quintessential example often used to explain fundamental probability concepts. Each roll is a singular, isolated event in terms of outcomes, especially when focusing on a fair six-sided die.
When you roll a die, it can land on any number from 1 through 6. Because there are six possible outcomes, the probability of rolling any specific number, such as a 1, is calculated by taking the number of favorable outcomes (just one for rolling a 1) divided by the total number of possible outcomes, which is 6. Therefore, the probability of rolling a particular number is 1/6.
Interestingly, each subsequent roll of a die doesn't affect the previous or next rolls. This property highlights the concept of independent trials in probability theory, making dice a perfect tool for illustrating basic principles of probability.