Problem 3 - Algebra 2 and Trigonometry

Question

What is the probability of getting a 2 on a single throw of a fair die?

Step-by-Step Solution

Answer

The probability of rolling a 2 on a fair die is \( \frac{1}{6} \).

Step 1: Understanding the Problem

We need to find the probability of rolling a 2 on a fair six-sided die. Each side of the die has an equal chance of landing face up.

Step 2: Identify Total Outcomes

A standard die has 6 faces, representing 6 equally likely outcomes (1, 2, 3, 4, 5, 6). Hence, the total number of possible outcomes when a die is rolled is 6.

Step 3: Identify Favorable Outcomes

The favorable outcome is rolling a 2. Since there is only one face of the die with a 2, there is exactly 1 favorable outcome.

Step 4: Calculate the Probability

Probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Thus, the probability of rolling a 2 is given by: \[ P( ext{rolling a 2}) = \frac{1}{6} \]

Key Concepts

Understanding Favorable Outcomes

Favorable outcomes in probability refer to the specific results that satisfy the condition you are interested in. Imagine rolling a die and aiming to get a 2. Here, the favorable outcome is rolling a 2. No other number will count for this particular question.
When dealing with dice, each number, from 1 to 6, can appear on a roll. But only one of those numbers is your favorable outcome.

• Favorable outcomes are what you want to happen—rolling a 2, for example.
• In our example with a die, there is only 1 favorable side.
• To identify favorable outcomes, pinpoint the scenarios that directly correspond to the question asked.

Calculating Total Outcomes

Total outcomes encompass all possible results of a given action. With a standard die, you have 6 faces, each representing an outcome when you roll the die: 1, 2, 3, 4, 5, and 6.

These outcomes are crucial for determining probabilities because they form the denominator when calculating the probability of any specific event occurring.

• Total outcomes include all potential numbers a die can land on.
• When rolling a fair die, there are always 6 possible results.
• Remember, without knowing the total outcomes, calculating probability correctly is impossible.

Understanding Dice Probability

Dice probability is a classic example in probability due to its simplicity and clarity. It is the likelihood of rolling a specific number on a fair six-sided die.
To calculate the probability of rolling a specific number, like a 2, divide the number of favorable outcomes (rolling a 2) by the total outcomes (6 numbers). This gives us \( P(\text{rolling a 2}) = \frac{1}{6} \).

• Probability is a fraction showing how likely something is based on all possible outcomes.
• It ranges from 0 (impossible event) to 1 (certain event).
• In dice probability, each number has an equal chance of appearing, which simplifies calculations.