Problem 50 - Algebra 2

Question

A die is rolled three times. Find each probability. \(P(\text { three } 1 \mathrm{s})\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{216}\).

Step 1: Understand the Problem

We need to find the probability of rolling a 1 on a six-sided die three consecutive times.

Step 2: Calculate the Probability for a Single Roll

For a fair six-sided die, the probability of rolling a 1 is \(\frac{1}{6}\).

Step 3: Use Independent Probability Formula

Since rolling a die is an independent event, the probability of rolling a 1 three times in a row is the product of the probabilities of each roll. Thus, we compute \(\left(\frac{1}{6}\right) \times \left(\frac{1}{6}\right) \times \left(\frac{1}{6}\right)\).

Step 4: Multiply the Probabilities

Multiply the probabilities from each roll: \(\left(\frac{1}{6}\right)^3 = \frac{1}{216}\).

Step 5: Result Interpretation

The probability of rolling three 1s in a row on a six-sided die is \(\frac{1}{216}\).

Key Concepts

Independent Events

When studying probability, understanding the concept of independent events is crucial. **Independent events** occur when the outcome of one event does not affect the outcome of another. For instance, when rolling a six-sided die, each roll is independent because the result of one toss does not impact the subsequent rolls.
Imagine flipping a coin and rolling a die simultaneously—these actions don't influence each other. This independence is essential when calculating probabilities in sequences of events.

• An independent event’s probability remains unchanged, regardless of other occurrences.
• This is fundamental in calculations involving repetitive independent events, such as the roll of a die.

Understanding this concept helps in recognizing situations where you can multiply probabilities to find combined outcomes.

Probability Calculation

Calculating the probability of independent events happening consecutively involves a straightforward approach. Let's break it down using the example of a six-sided die.
The first step is determining the likelihood of a single event. For a six-sided die, the probability of rolling any specific number (say, 1) is expressed as:

• Single Roll Probability: \( \frac{1}{6} \)

Once you've got the probability of a single event, the next step is applying the formula for independent events:
When events are independent, their combined probability is the product of their individual probabilities.
For our task of rolling a 1 three times in a row, the calculation would be:

• Combined Probability: \( \left( \frac{1}{6} \right) \times \left( \frac{1}{6} \right) \times \left( \frac{1}{6} \right) \)

Thus, the result is \( \frac{1}{216} \), showing that consecutive probabilities require multiplication due to their independence.

Six-Sided Die

A six-sided die is a standard tool in probability exercises. Each face of the die has an equal chance of landing face up. This makes it a balanced and fair option for examining random events.
The principal characteristics of a six-sided die are:

• Six faces, each marked with a number from 1 to 6.
• An equal probability of \( \frac{1}{6} \) for landing on any particular number.

Using a six-sided die is a classic example in probability because each number’s chance is equal. When this die is rolled, the simplicity of its design allows anyone to calculate probabilities for sequences and patterns effectively.
This helps demonstrate core statistical concepts, making it a vital tool for learning about randomness and independent events.