Question

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

Short answer

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

Step-by-step solution

Identify the number of possible outcomes

There are 6 sides on a fair die, numbered 1 to 6. So, there are 6 possible simple events which we denote by {1, 2, 3, 4, 5, 6}.

Assign probabilities to the simple events

Since the die is fair, every side has an equal chance of occurring. Therefore, the probability of each simple event is \(\frac{1}{6}\). So, we have the following probabilities: P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = \(\frac{1}{6}\).

Define the event \(C\)

The event \(C\) is defined as observing a number greater than 2, i.e., {3, 4, 5, 6}.

Calculate the probability of the event \(C\)

To find the probability of event \(C\), add the probabilities of the simple events that make up \(C\): {3, 4, 5, 6}. We have: P(C) = P(3) + P(4) + P(5) + P(6) = \(\frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{4}{6}\).

Simplify the probability of the event \(C\)

Simplify the fraction \(\frac{4}{6}\) : P(C) = \(\frac{2}{3}\). The probability of observing a number greater than 2 when tossing a single fair die is \(\frac{2}{3}\).

Key concepts

Understanding Fair Die Probability

When we talk about a 'fair die', we refer to an ideal dice that has equal chances of landing on any of its faces when thrown. It's essential in understanding probability, as it provides a basis for an unbiased outcome.

A standard fair die has six faces, each marked with a different number of dots from 1 to 6. Since it is fair, each of these numbers has an equal chance of coming up, which is the core concept behind the 'fair die probability'.

In mathematical terms, this equal probability is represented as \( \frac{1}{6} \) for each face, because there are six possible outcomes and each is equally likely. Thus, when you roll a fair die, the probability of rolling any single number is always one-sixth.

Calculating Event Probability

The likelihood of an event happening in a probability experiment is referred to as the 'event probability'. When we calculate this probability, we're essentially asking, 'What are the chances of a certain event occurring?'

Event probability is calculated by dividing the number of ways an event can occur by the total number of possible outcomes. In the case of a fair die, if we want to calculate an event probability, we list all the outcomes that fulfill our event condition and then add their probabilities together.

For example, as in the given exercise, if we are looking for the probability of rolling a number greater than 2 (event C), we only consider the favorable outcomes: 3, 4, 5, and 6. We then sum the probabilities for these outcomes: \( P(C) = P(3) + P(4) + P(5) + P(6) \) which gives us a result of \( \frac{2}{3} \) after simplification.

Simple Events in Probability

A 'simple event' in probability is an event that cannot be broken down into smaller parts. In other words, it's an outcome that's completely independent from any other outcome and has a single possibility.

In a fair die throw, each number rolling out is considered a simple event because it stands alone and does not depend on other rolls. The outcomes {1}, {2}, {3}, {4}, {5}, or {6} are all examples of simple events - each with a probability of \( \frac{1}{6} \) as provided in the exercise.

Understanding simple events is crucial since more complex events can often be expressed as combinations of simple events. This idea helps when calculating the overall event probability, making it easier to manage and understand the probability of various outcomes, especially when dealing with compound events that encompass multiple simple events.