Question

A single fair die is tossed. Assign probabilities to the simple events and calculate the probabilities. \(A:\) Observe a 2

Short answer

Answer: The probability of observing a 2 when tossing a fair die is 1/6.

Step-by-step solution

Assign probabilities to simple events

: There are 6 possible outcomes when tossing a fair die, each representing the face number showing up (1, 2, 3, 4, 5, and 6). Since it is a fair die, all outcomes have an equal chance (1/6) of occurring. Assign the probability of \(1/6\) to each of the simple events.

Determine the favorable condition for event A

: Event A is to observe a 2 on the die. There is only one favorable outcome for this event: the die shows a face with the number 2.

Calculate the probability of event A happening

: The probability of event A (observing a 2 on the die) is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In this case, there is only one favorable outcome (2), and there are 6 possible outcomes (1, 2, 3, 4, 5, and 6). So, the probability of event A happening is: \(\text{P(A)} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{6}\) Therefore, the probability of observing a 2 on the die is \(\frac{1}{6}\).

Key concepts

Simple Events

In probability, a simple event refers to an outcome that cannot be broken down further. It represents one specific result from a random experiment.
For instance, when you roll a die, observing a 2 is considered a simple event.
Each number on the die from 1 to 6 serves as its own distinct event.

• A simple event occurs at the most basic level.
• In a fair die toss, each side (1, 2, 3, 4, 5, 6) is a simple event.

Understanding simple events is crucial as they form the foundation for calculating probabilities in more complex scenarios.

Fair Die

A fair die is a theoretical concept where each face of the die has an equal chance of landing face up. This means no bias exists in how the die is weighted or shaped.
In practice, a fair die ensures randomness and unpredictability in the results of a dice roll.

• A standard die has six sides, numbered 1 through 6.
• Each side has an equal probability, i.e., \(\frac{1}{6}\), of occurring.

This equality ensures a balanced and impartial experiment, making the study of probabilities both reliable and fair.

Calculating Probabilities

Calculating probabilities involves determining how likely an event is to occur. The general formula is to divide the number of favorable outcomes by the total number of possible outcomes.
For a single die toss, this calculation is straightforward.

Formula

\[\text{P(A)} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]

Example

Consider the event of observing a 2 on a die.

• Favorable outcome: One (the number 2).
• Total possible outcomes: Six (1, 2, 3, 4, 5, 6).

Applying the formula: \text{P(A)} = \frac{1}{6}\.
Thus, the probability is \(\frac{1}{6}\). This method applies to any simple event in a similar setup.

Sample Space

The sample space is the set of all possible outcomes in a probability experiment. For a single die roll, the sample space includes every potential result of the roll.
It forms the basis for defining events and assigning probabilities.

• Sample space for a die: {1, 2, 3, 4, 5, 6}
• Every number in the set is equally likely for a fair die.

This concept helps organize how different outcomes relate to each other and simplifies the calculation of probabilities by providing a clear picture of all possibilities.