14.8 Chapter summary (EMA85)

Presentation: 2GX2

• An experiment refers to an uncertain process.

• An outcome of an experiment is a single result of that experiment.

• The sample space of an experiment is the set of all possible outcomes of that experiment. The sample space is denoted with the symbol \(S\) and the size of the sample space (the total number of possible outcomes) is denoted with \(n\left(S\right)\).

• An event is a specific set of outcomes of an experiment that you are interested in. An event is denoted with the letter \(E\) and the number of outcomes in the event with \(n\left(E\right)\).

• A probability is a real number between \(\text{0}\) and \(\text{1}\) that describes how likely it is that an event will occur.

  • A probability of \(\text{0}\) means that an event will never occur.

  • A probability of \(\text{1}\) means that an event will always occur.

  • A probability of \(\text{0,5}\) means that an event will occur half the time, or \(\text{1}\) time out of every \(\text{2}\).

• A probability can also be written as a percentage or as a fraction.

• When all of the possible outcomes of an experiment have an equal chance of occurring, we can compute the exact theoretical probability of an event. The probability of an event is the ratio between the number of outcomes in the event set and the number of possible outcomes in the sample space.

  \[P(E) = \frac{n(E)}{n(S)}\]

• The relative frequency of an event is defined as the number of times that the event occurs during experimental trials, divided by the total number of trials conducted.

  \[f = \frac{\text{number of positive trials}}{\text{number of trials}} = \frac{p}{n}\]

• The union of two sets is a new set that contains all of the elements that are in at least one of the two sets. The union is written as \(A \cup B\) or \(A \text{ or } B\).

• The intersection of two sets is a new set that contains all of the elements that are in both sets. The intersection is written as \(A \cap B\) or \(A \text{ and } B\).

• The probability of observing an outcome from the sample space is 1: \(P\left(S\right) = 1\).

• The probability of the union of two events is calculated using: \(P\left(A \cup B\right) = P\left(A\right) + P\left(B\right) - P\left(A \cap B\right)\).

• Mutually exclusive events are two events that cannot occur at the same time. Whenever an outcome of an experiment is in the first event, it can not also be in the second event.

• The complement of a set, \(A\), is a different set that contains all of the elements that are not in \(A\). We write the complement of \(A\) as \(A'\) or “\(\text{not }\left(A\right)\)”.

• Complementary events are mutually exclusive: \(A \cap A' = \varnothing\).

• Complementary events cover the sample space: \(A \cup A' = S\)

• Probabilities of complementary events sum to \(\text{1}\): \(P\left(A\right) + P\left(A'\right) = P\left(A\cup A'\right) = P\left(S\right) = 1\).