Question

State whether the two events (A and B) described are disjoint, independent, and/or complements. (It is possible that the two events fall into more than one of the three categories, or none of them.) South Africa plays Australia for the championship in the Rugby World Cup. Let \(\mathrm{A}\) be the event that Australia wins and \(\mathrm{B}\) be the event that South Africa wins. (The game cannot end in a tie.)

Short answer

Event A and Event B are disjoint and complementary, but not independent.

Step-by-step solution

- Identify disjoint events

Two events are disjoint if they can't happen at the same time. Here, Australia and South Africa can't both win the game at the same time. That means these events are disjoint.

- Identify independent events

Two events are independent if the occurrence of one doesn't affect the likelihood of the other. Since there can be only one winner, if Australia wins (Event A), South Africa cannot win (Event B). So, Event A and Event B are not independent.

- Identify Complement Events

Two events are said to be complementary if the occurrence of one implies the non-occurrence of the other. Here, if Australia wins (Event A) it means South Africa (Event B) doesn't win and vice versa. Hence, these two events are complementary.

Key concepts

Disjoint Events

Disjoint events are like oil and water; they don't mix. In probability, when we talk about disjoint events, we mean occurrences that cannot happen at the same time. Think of it as being exclusive. For example, consider flipping a coin. You can get heads or tails, but not both at once. Similarly, when a rugby match between Australia and South Africa is played without the possibility of a draw, only one team can be victor. If Australia wins, South Africa can't, making these two outcomes disjoint events.

Disjoint events always have one thing in common: their intersection (the probability of both events happening together) is zero. Mathematically, we represent it as: \( P(A \cap B) = 0 \) where \( A \) and \( B \) are two disjoint events. Understanding this concept helps you predict outcomes when you're dealing with exclusive scenarios in games, experiments, or real-life situations.

Independent Events

Now imagine you have two light switches. Turning one on or off has no bearing on whether the other is on or off. Independent events work much the same. In probability, two events are independent if the occurrence (or non-occurrence) of one has no effect on the other. This concept is fundamental in scenarios where multiple outcomes don't interfere with each other.

To check for independence, we look at the probabilities: if \( P(A \text{ given } B) = P(A) \) and \( P(B \text{ given } A) = P(B) \) then \( A \) and \( B \) are independent. But remember, during the Rugby World Cup game between Australia and South Africa, one team's victory completely determines the other team's defeat; hence, their chances of winning are not independent. Knowing helping you to understand when one occurrence doesn't necessarily predict another.

Complement Events

Complement events behave like two sides of the same coin. In probability, the complement of event \( A \) is the event that \( A \) does not occur, often denoted by \( A^c \). Think of it as a 'totally or nothing' situation. When it comes to our Rugby game, Australia winning is the complement of South Africa not winning, and vice versa. One complements the other to make up all possible outcomes.

Here's the trick: The probabilities of an event and its complement always add up to 1 (or 100%). So, if you know the probability of Australia winning, \( P(A) \), the probability of them not winning (South Africa winning or event \( B \)) is \( 1 - P(A) \). This is incredibly useful for calculating probabilities quickly, especially in 'either-or' scenarios. Complement events give us a full picture of the possible outcomes in a given situation.