Question

What is the probability of rolling a 6 on two consecutive rolls of a fair six- sided die?

Short answer

The probability of rolling a 6 on two consecutive rolls of a fair six-sided die is \( \frac{1}{36} \).

Step-by-step solution

Understand the probability of a single event

A fair six-sided die has 6 faces, numbered 1 to 6. The probability of rolling a 6 on a single roll is \( \frac{1}{6} \).

Identify the event of interest

The event of interest is rolling a 6 on two consecutive rolls of the die.

Calculate the combined probability

Since the two rolls are independent events, the combined probability is the product of the individual probabilities: \[ P(\text{6 on first roll}) \times P(\text{6 on second roll}) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \]

Key concepts

Independent Events

In probability, independent events are those events where the outcome of one event doesn't affect the outcome of another. Take rolling a fair six-sided die, for instance. If you roll a 6 on the first roll, it doesn't change the probability of rolling a 6 on the second roll.
Simply put, each roll of the die is an independent event.
This is important when calculating combined probabilities because it allows us to multiply the probabilities of individual events to get the combined probability.

Combined Probability

When dealing with independent events, you can find the combined probability by multiplying the probability of each event.
For example, if you want to know the probability of rolling a 6 on two consecutive rolls of a fair six-sided die, you multiply the probability of rolling a 6 on the first roll by the probability of rolling a 6 on the second roll.
The calculation is as follows: \( P(\text{6 on first roll}) \times P(\text{6 on second roll}) \). Since each single roll has a probability of \( \frac{1}{6} \), the combined probability is \( \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \).

Fair Six-Sided Die

A fair six-sided die is a cube with faces numbered from 1 to 6. When we say a die is 'fair,' we mean that each face has an equal chance of landing face up.
That means when you roll the die, each number from 1 to 6 has a probability of \( \frac{1}{6} \). This equal probability is a key factor in calculating probabilities involving dice.
Because the die is fair, each roll is a random and independent event. This makes it simpler to calculate the combined probabilities of multiple rolls.

Basic Probability

Probability is the measure of how likely an event is to occur. It ranges from 0 to 1, where 0 means the event cannot happen and 1 means it will definitely happen.
The probability of any single event occurring is calculated as the number of favorable outcomes divided by the total number of possible outcomes.
For example, if you want to calculate the probability of rolling a 6 on a fair six-sided die, there is 1 favorable outcome (rolling a 6) and 6 possible outcomes (rolling a 1, 2, 3, 4, 5, or 6).
So, the probability is \( \frac{1}{6} \).