Question

WRITING Are the events \(A\) and \(\bar{A}\) disjoint? Explain. Then give an example of a real-life event and its complement.

Short answer

Yes, events \(A\) and \( \bar{A} \) are disjoint, because they are complementary, meaning if one event occurs, the other cannot. An example would be the event of 'It will rain tomorrow' and its complement 'It will not rain tomorrow.'

Step-by-step solution

Defining concepts

In probability theory, event \(A\) and its complement \( \bar{A} \) refer to two mutually exclusive outcomes. That is, if event \(A\) happens, event \( \bar{A} \) cannot occur, and vice versa. Therefore, by definition, these two events are disjoint because they cannot happen at the same time.

Examples of Complement Events

Consider a real-life situation where event \(A\) is 'It will rain tomorrow.' The complement, \( \bar{A} \), would be 'It will not rain tomorrow.' These two events are clearly disjoint since at any given day, it can either rain or not rain, but not both.

Interpretation

In general, for any event \(A\), \(A\) and \( \bar{A} \) are disjoint, as they represent all the possible outcomes in the sample space, and they don't share any common outcome.

Key concepts

Probability Theory

At the heart of understanding random events is probability theory, a branch of mathematics that deals with the likelihood of different outcomes. Whether it's rolling a dice, picking a card from a deck, or forecasting the weather, probability theory provides a framework for making predictions about events when there's uncertainty involved.

Central to probability theory is the idea of a sample space, which is the set of all possible outcomes of a random experiment. For instance, the sample space for a coin toss is {Heads, Tails}. Probability theory also uses events to represent subsets of the sample space that we are interested in. The likelihood of an event occurring is quantified as a number between 0 and 1, known as the probability of the event. A probability of 0 means the event will not occur, while a probability of 1 guarantees that the event will happen.

To put this into practice, consider tossing a fair coin. The probability of getting heads is 0.5, as is the probability of getting tails. Probability theory helps not just in calculating these probabilities, but also in understanding the relationships between different events, like whether they're independent or dependent, mutually exclusive or not.

Complement of an Event

In probability, the complement of an event encompasses all outcomes in the sample space that are not in the event. This concept is crucial when determining probabilities because every event's probability and its complement's probability always add up to 1.

For example, if we define event A as drawing a red card from a standard deck of cards, the complement of A (denoted as \( \bar{A} \)) includes all outcomes that are not red cards, which would be the black cards. So, if the probability of drawing a red card (event A) is 0.5, the probability of not drawing a red card (complement of A) is also 0.5. The probabilities are complementary; one fills in where the other leaves off, covering the entire sample space without overlap. This is why understanding the complement is so useful; it allows for a thorough and complete approach to probability calculations where considering all possibilities is imperative.

Sample Space

The sample space is the cornerstone of any probability problem. It is defined as the set of all possible outcomes of a probabilistic experiment. Having a well-defined sample space is essential because it ensures that all the events within the realm of possibility are accounted for.

For instance, if the experiment is rolling a single six-sided die, the sample space is \( S = \{1, 2, 3, 4, 5, 6\} \), which lists out each numeral that could land face up after the toss. When analyzing events within this sample space, it's straightforward to delineate which outcomes belong to an event, which are disjoint, and which make up the complement of an event. Not to mention, a clearly defined sample space is fundamental when considering complex events, such as the event of rolling an even number, which would correspond to the outcomes {2, 4, 6} within the declared sample space.