Question

A single fair die is tossed. Assign probabilities to the simple events and calculate the probabilities. \(D:\) Observe a number less than 5

Short answer

Answer: The probability of observing a number less than 5 when tossing a single fair die is \(\frac{2}{3}\).

Step-by-step solution

Identify the Simple Events

A single fair die has 6 faces, numbered from 1 to 6. Therefore, when it is tossed, there are 6 simple events: {1, 2, 3, 4, 5, 6}.

Assign Probabilities to Simple Events

Since the die is fair, the probability of each number appearing on a roll is \(\frac{1}{6}\). So we assign the equal probability, \(\frac{1}{6}\), to each of the simple events: {1, 2, 3, 4, 5, 6}.

Identify the Event of Interest

The event \(D\) is where we observe a number less than 5. Using the simple events, we can form the event \(D\) as: \(D = \{1, 2, 3, 4\}\).

Calculate the Probability of Event D

The probability of a number less than 5 is the sum of the probabilities of the simple events in the set \(D\). So, we will add the probabilities we assigned to each of the simple events in \(D\): \(P(D) = P(1) + P(2) + P(3) + P(4) = \frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6} = \frac{4}{6} =\frac{2}{3}\). The probability of observing a number less than 5 when tossing a single fair die is \(\frac{2}{3}\).

Key concepts

Probability Theory

Probability theory is a branch of mathematics concerned with analyzing random events. The foundation of probability theory involves understanding outcomes, events, and the likelihood of these events occurring in various scenarios. An essential part of probability theory is the concept of simple events.

Simple events are outcomes that cannot be broken down into simpler components. For example, rolling a certain number on a fair die is considered a simple event because it is a singular, undividable outcome. Probability theory allows us to assign a numeric value, representing the chance that a particular event will take place, which is expressed as a number between 0 and 1. A 0 means the event is impossible, while a 1 signifies that the event is certain to occur.

Understanding the basics of probability theory is crucial because it applies to a range of fields, from science and engineering to economics and social sciences, providing us with a toolkit to handle uncertainty and make informed predictions.

Fair Die Probability

A fair die is a classic example used to explain basic probability. The 'fairness' of the die means that each of its six faces (typically numbered from 1 to 6) has an equal chance of landing face up when the die is rolled. Therefore, the probability associated with any single number coming up is the same for all numbers.

Since there are 6 faces, and each is equally likely to occur, the probability of any one face turning up on a roll is \(\frac{1}{6}\). This is assuming the die is not biased in any way and that each roll is independent of the others. Recognizing that the fair die is a perfect model for illustrating uniform probability distribution helps us not only in understanding games of chance like board games or gambling but also in creating statistical models and simulations.

Simple Events in Probability

Simple events in probability refer to outcomes that are both mutually exclusive and exhaustive. Mutually exclusive events cannot occur at the same time, and exhaustive means that the set of all simple events covers all possible outcomes.

In the context of rolling a fair die, simple events are the individual rolls resulting in just one number showing up each time. For example, rolling a 2 and a 4 are both simple events and cannot happen simultaneously when rolling a single die. The sum of the probabilities of all simple events must equal 1, which reflects the certainty that one of these events will occur on a roll. When calculating probabilities like in the die example, we combine the probabilities of the relevant simple events to find the likelihood of more complex outcomes.