91Ó°ÊÓ

In probability theory, a complementary event is essentially the "opposite" of a given event. When we talk about the event \( A \) which is rolling a die and getting a number less than 3, the complementary event \( A' \) would be rolling anything that is not less than 3. This means the outcomes in \( A' \) are 3, 4, 5, and 6.
It is important to understand that whenever you have an event, its complement will cover all other possible outcomes that are not part of the original event. This is useful because the probability of all possible events (including the event and its complement) always sums up to 1.
In formula terms:

• \( P(A) + P(A') = 1 \)

This means you can quickly find the probability of the complementary event if you know the probability of the original event, and vice-versa.

A standard die is a cube with six faces, numbered from 1 through 6. When you roll a die, each face has an equal chance of landing face up, leading to six possible outcomes. This includes the numbers:

• 1
• 2
• 3
• 4
• 5
• 6

Each of these outcomes has a probability of occurring which is \( \frac{1}{6} \) since the die is fair. Understanding these equal probabilities is crucial when determining the likelihood of certain events during a die roll.
For instance, if an event \( A \) is "rolling a number less than 3," the outcomes that satisfy this condition are 1 and 2.

The sample space is a fundamental concept in probability. It refers to the set of all possible outcomes of a particular experiment or trial. In the scenario of rolling a single die, the sample space \( S \) is:

• \( S = \{1, 2, 3, 4, 5, 6\} \)

Each element in this sample space represents a possible outcome from the die roll.
Understanding the sample space is crucial because it is the foundation upon which events are defined. When calculating probabilities, the denominator is often the size of the sample space since it represents the total number of possible outcomes.
For example, in finding the probability of an event \( A \), we express it as \( \frac{n(A)}{n(S)} \), where \( n(S) \) is the number of elements in the sample space.

The calculation of probability is a straightforward yet essential concept in understanding any probability problem. It involves dividing the number of favorable outcomes by the total number of possible outcomes.
Probability is represented as:

• \( P(A) = \frac{n(A)}{n(S)} \)

Here, \( n(A) \) is the number of favorable outcomes for the event \( A \), and \( n(S) \) is the total number of outcomes in the sample space.
Taking the example of the event \( A \) which is rolling a number less than 3, there are 2 favorable outcomes (rolling a 1 or 2). Out of 6 possible outcomes from the die, the probability is \( \frac{2}{6} = \frac{1}{3} \).
Similarly, for the complementary event \( A' \), there are 4 favorable outcomes (numbers 3 through 6). The corresponding calculation is \( \frac{4}{6} = \frac{2}{3} \).