10.6 Summary (EMBJY)

• Terminology:

  • Outcome: a single observation of an experiment.
  • Sample space of an experiment: the set of all possible outcomes of the experiment.
  • Event: a set of outcomes of an experiment.
  • Probability of an event: a real number between \(\text{0}\) and \(\text{1}\) that describes how likely it is that the event will occur.
  • Relative frequency of an event: the number of times that the event occurs during experimental trials, divided by the total number of trials conducted.
  • Union of events: the set of all outcomes that occur in at least one of the events, written as “\(A \text{ or } B\)”.
  • Intersection of events: the set of all outcomes that occur in all of the events, written as “\(A \text{ and } B\)”.
  • Mutually exclusive events: events with no outcomes in common, that is \((A \text{ and } B) = \emptyset\).
  • Complementary events: two mutually exclusive events that together contain all the outcomes in the sample space. We write the complement as “\(\text{not } A\)”.
  • Independent events: two events where knowing the outcome of one event does not affect the probability of the other event. Events are independent if and only if \(P(A\text{ and }B) = P(A) \times P(B)\).

• Identities:

  • The addition rule: \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\)
  • The addition rule for \(\text{2}\) mutually exclusive events: \(P(A \text{ or } B) = P(A) + P(B)\)
  • The complementary rule: \(P(\text{not } A) = 1 - P(A)\)

• A Venn diagram is a visual tool used to show how events overlap. Each region in a Venn diagram represents an event and could contain either the outcomes in the event, the number of outcomes in the event or the probability of the event.

• A tree diagram is a visual tool that helps with computing probabilities for dependent events. The outcomes of each event are shown along with the probability of each outcome. For each event that depends on a previous event, we go one level deeper into the tree. To compute the probability of some combination of outcomes, we

  • find all the paths that contain the outcome of interest;
  • multiply the probabilities along each path;
  • add the probabilities between different paths.

• A two-way contingency table is a tool for organising data, especially when we want to determine whether two events, each with only two outcomes, are dependent or independent. The counts for each possible combination of outcomes are entered into the table, along with the totals of each row and column.