# Fundamentals of probability part III

03/03/2020

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## Outline

• The following topics will be covered in this lecture:
• 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. 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 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 $$\overline{A}$$.