Fundamentals of probability part III

03/03/2020

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Outline

  • The following topics will be covered in this lecture:
    • Review of addition rule
    • Review of conditional probability
    • Review of the multiplication rule
    • Review of independence
    • Further notes on independence
    • Further uses of conditional probability
    • Bayes' Law

Compound events

  • We will often be concerned not with one event \( A \) but some combination of some event \( A \) and some event \( B \).
  • Compound event – formally we define a compound event as any event combining two or more simple events.
  • There are two key operations joining events
    1. “OR” – in mathematics we refer to “or” as a non-exclusive “or”.
      • The meaning of this for “\( A \) or \( B \)” is – event \( A \) occurs, event \( B \) occurs, or both events \( A \) and \( B \) occur.
      • We will not consider the exclusive “or”, i.e. either event \( A \) occurs, or event \( B \) occurs, but not both.
    2. “AND” – in mathematics we refer to “and” in an exclusive sense.
      • The meaning of this for “\( A \) and \( B \)” is – both event \( A \) and event \( B \) occurs.
  • The operations “and” and “or” join events together in a way that we can compute the probability of the joint events.

Addition rule

Venn diagram of events \( A \) and \( B \) with nontrivial intersection.

Courtesy of Bin im Garten CC via Wikimedia Commons

  • Suppose we want to compute the probability of two events \( A \) and \( B \) joined by the compound operation “or”.
  • We read the statement, \[ P(A \text{ or } B) \] as he probability of event:
    • \( A \) occuring,
    • event \( B \) occuring, or
    • both \( A \) and \( B \) ocurring.
  • Intuitively, we can express the probability in terms of all the ways \( A \) can occur and all the ways \( B \) can occur, if we don’t double count.
  • Let all the ways that \( A \) can occur be represented by the red circle to the left.
  • Let all the ways that \( B \) can occur be represented by the dashed circle to the left.
  • If there is any overlap between events \( A \) and \( B \) so that they can occur simultaneously, \( P(A) +P(B) \) counts the cases where \( A \) and \( B \) both occur twice.
  • Therefore, the addition rule for compound events is given as, \[ P(A\text{ or }B) = P(A) + P(B) - P(A\text{ and }B) \]

Complementary events

Venn diagram of events \( A \) and \( B \) with nontrivial intersection.

Courtesy of Bin im Garten CC via Wikimedia Commons

  • A special case of the addition rule comes up when the events \( A \) and \( B \) are disjoint or mutually exclusive.
    • When \( A \) and \( B \) are disjoint, this means that there is no overlap between these events and they will never occur simultaneously.
    • In this case \( P(A \text{ and } B) = 0 \), so the addition rule becomes, \[ \begin{align} P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) &= P(A) + P(B) \end{align} \]
  • Recall the complement of \( A \), denoted \( \overline{A} \) is the event where \( A \) does not occur.
    • By definition, \( A \) and \( \overline{A} \) are disjoint because \( A \) will not both occur and not occur simultaneously.
  • However, complementary events make up all possible outcomes – \( A \) will either occur or not occur, so that we are certain about the outcome of \( P(A \text{ or } \overline{A}) \).
    • That is, we know by definition that \[ P(A \text{ or } \overline{A}) = 1 \]
  • Using the above fact, along with the disjointness of \( A \) and \( \overline{A} \) with the addition rule, \[ 1= P\left(A\text{ or }\overline{A}\right) = P(A) + P\left(\overline{A}\right). \] for any event \( A \) and its complement \( \overline{A} \).

Multiplication rule