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. At the same time, Poland plays Russia for the World Team Chess Championship. Let \(\mathrm{A}\) be the event that Australia wins their rugby match and \(\mathrm{B}\) be the event that Poland wins their chess match.

Short answer

The two events, A (Australia wins their rugby match) and B (Poland wins their chess match), are independent but not disjoint or complementary.

Step-by-step solution

Determining if events are Disjoint

Disjoint events cannot occur simultaneously. In this context, event A is Australia winning their rugby match and event B is Poland winning their chess match. These two events involve different countries and different sports. The outcome of event A does not prevent event B from happening, and vice versa. Hence, events A and B are not disjoint.

Determining if events are Independent

Independent events are those where the outcome of one event does not affect the outcome of the other. Here, whether Australia wins their rugby match or not, it doesn't influence the outcome of the chess match between Poland and Russia. Thus, event A happening (or not happening) does not change the probability of event B happening (or not happening). Therefore, events A and B are independent.

Determining if events are Complementary

Complementary events together cover all possible outcomes of a particular experiment. In our case, there are no such outcomes that justify the condition of complementarity for the events A and B. The win of Australia in rugby and Poland in chess do not cover all possible outcomes of the sporting events described and happening of one does not imply the non-happening of the other. Therefore, A and B are not complementary events.

Key concepts

Disjoint Events

When discussing probabilities in the area of statistics, a concept that often comes up is that of disjoint events. Disjoint events, also known as mutually exclusive events, are events that cannot occur at the same time. An easy way to remember this is to think of 'disjoint' as 'does not join together'. For example, consider rolling a six-sided die. The event of rolling a two (Event A) and rolling a five (Event B) are disjoint events because you can't roll a two and a five on the same single die roll.

In the exercise involving sports events, determining whether the events are disjoint requires examining if they can happen simultaneously. Since Australia winning a Rugby match and Poland winning a Chess match are occurrences in different sports with different participants, the results of one do not interfere with the possibility of the other occurring. They are separate occurrences, hence, not disjoint events because the concept of 'disjoint' does not apply to events in different, unrelated experiments or scenarios.

Independent Events

Understanding the independence of events is another foundational aspect of probability. Independent events are defined by the characteristic that the outcome of one event does not affect the outcome of the other. To put it simply, they don't influence each other. A classic example of independent events is flipping a coin and rolling a die. Whether you flip heads or tails does not change the chances of rolling a six.

In the textbook exercise, the independence of the events is assessed based on whether the outcome of the rugby match between South Africa and Australia influences the chess match between Poland and Russia, and vice versa. Given that one is a rugby match and the other a chess match, occurring separately and with no shared factors, the events are deemed independent. Regardless of who wins the rugby match, it doesn't change the likelihood of Poland winning or losing their chess match. Students often enhance their understanding by noting that in independent events, knowing the outcome of one has no bearing on the probability of the other.

Complementary Events

Another crucial concept in probability is that of complementary events. Complementary events are defined...

... as two mutually exclusive events whose probabilities add up to one, meaning together they represent all possible outcomes of a specific experiment. Think of it like a coin flip: if Event A is flipping heads, then Event B, flipping tails, is the complement of A because there are no other outcomes for the coin flip. When you sum the probabilities of flipping heads and flipping tails, you get one, which is the certainty of flipping a coin and getting some result.

In the case of our sports championships scenario, we're looking at four different teams in two different sports, so we can't define any pair of these teams winning their respective matches as complementary. Australia's victory in rugby or Poland's victory in chess doesn't say anything about the other match's outcome, and they certainly don't represent all potential outcomes when paired. Therefore, in the context of this exercise, the events are not complements.