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Chapter 13: Problem 48

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

Short Answer

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

Step by step solution

01

Understanding the Problem

We need to find the probability of rolling a die twice and getting a 'one' both times. Each roll is an independent event.
02

Determine Probability for One Roll

The probability of rolling a 'one' on a single die roll is \( \frac{1}{6} \) since there are six sides and only one "one" on a die.
03

Calculate Probability for Both Rolls

Since the two rolls are independent, the probability of both events occurring is the product of their individual probabilities. Thus, the probability of rolling 'one' on both rolls is \( \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
In probability theory, independent events are events whose outcomes do not affect each other. This concept is crucial when dealing with multiple events, like rolling dice. For instance, when you roll a die twice, the outcome of the first roll does not influence the second roll.

To visualize:
  • The result of the first die roll is separate from the second.
  • Each die roll is treated as a standalone event.
  • Any event occurring in the first roll does not change the likelihood of any event in the second roll.
Recognizing independent events helps in calculating probabilities because we can use the multiplication rule, knowing the events do not interfere with each other.
Rolling Dice
Rolling a die is a simple yet fascinating experiment in probability. Each die has six faces, numbered from 1 to 6. When you roll a die, there are six possible outcomes. Therefore, the probability of any single number showing up on a roll is uniform, or equally likely. This means each outcome has the same chance.
  • For a single die roll, the probability of each face (like rolling a 'one') is \( \frac{1}{6} \).
  • Each number has an equal chance of appearing.
  • Repeated rolls won't affect the probability of each face showing.
This fairness means that when you roll multiple times, each roll starts fresh with the same chances, embodying the idea of independent events.
Multiplication Rule for Probabilities
The multiplication rule for probabilities is a tool used to find the probability of two or more independent events all occurring. When events do not influence each other, like rolling a die, you can multiply their individual probabilities to find the joint probability.
  • It assumes events are independent.
  • You multiply the probability of each event.
For example, the probability of rolling a 'one' on a die is \( \frac{1}{6} \). If a die is rolled twice, and the result of one roll doesn't affect the other, the probability of getting a 'one' both times is \( \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \).

This law simplifies the calculation of complex probabilities by breaking them down into manageable parts and ensuring accuracy when events are independent.

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Most popular questions from this chapter

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