Question

You roll a fair die five times, and you get a 6 each time. What is the probability that you get a 6 on the next roll?

Short answer

The probability is \( \frac{1}{6} \).

Step-by-step solution

Understanding the Problem

We are rolling a fair die, which means each face of the die has an equal chance to land face up, independent of previous rolls. We are interested in finding the probability of rolling a 6 on the next roll.

Determine the Probability of a Single Event

For a fair 6-sided die, each face (including the one with 6) has an equal likelihood of appearing on any roll. This probability is determined by dividing 1 by the number of sides on the die. Therefore, the probability of rolling a 6 on a single roll is \[P( ext{Rolling a 6}) = \frac{1}{6}\]

Analyzing the Independence of Events

The outcome of each roll of the die is independent of the previous rolls. This means that the probability of getting a 6 on any given roll remains the same, regardless of past outcomes.

Conclude the Probability for a Single Roll

Since the die rolls are independent, the probability of rolling a 6 on the next roll is simply the probability of getting a 6 on any standard roll of a die.

Key concepts

Independent Events

In probability theory, independent events are those whose outcomes do not affect each other. Each roll of a die is an independent event because the result of one roll doesn't impact the result of the next roll. For instance, if you roll a fair die once, you might get a 6, and if you roll it again, the chance of getting a 6 remains the same — even if you keep rolling sixes repeatedly.

This independence is important because it assures that the probability of any specific outcome remains constant over each trial. No matter how many sixes have rolled in the past, the die doesn't "remember" previous results. Hence, each roll is a fresh event with consistent odds.

Key aspects of independent events include:

• The likelihood of outcomes in one event remains unchanged by other events.
• This independence allows predictions to be based on individual probabilities of each event.
• Each roll of a die, coin flip, or similar mechanism is individual and separate from others.

Fair Die

A fair die is a theoretical concept in probability representing an ideal condition where all outcomes are equally likely. In practical terms, a fair die is six-sided, with numbers one through six, providing a uniform chance of landing on any specific side.

The term 'fair' indicates balance and symmetry, meaning there is no bias or unequal weight distribution in the die. Thus, it follows that the probability of landing on any face, including the face numbered 6, is evenly split among all six faces. This means each number (such as rolling a 6) has an equal chance of appearing.

When rolling a fair die, think about:

• Each number has a 1 in 6 chance (\[\frac{1}{6}\]
• No face is favored over another due to design or conditions.
• It models a perfectly random and unbiased scenario often used in teaching probability principles.

Single Event Probability

The concept of single event probability involves finding the chance of a specific outcome in a single trial of an experiment. For a fair die, determining the single event probability means figuring out what's the likelihood of, say, rolling a 6 on one particular roll.

Each face of a standard die is equally probable to land on top. Therefore, the probability for rolling any chosen number, including 6, is calculated by dividing 1 by the number of sides on the die: \[ P(\text{Rolling a 6}) = \frac{1}{6} \]

Single event probability helps in understanding how likely an occurrence is and provides a foundation for more complex probability calculations. Some points to consider:

• Probabilities are expressed as a fraction, decimal, or percentage.
• This type of probability applies to single occurrences, making it a fundamental building block of more intricate scenarios.
• Understanding this helps in deciphering unpredictability in random experiments and events.