Probability

Complementary probabilities

The sum of probabilities must add to one, so the sum of the probabilities for an event occuring and for not occurring must add to one. We therefore state that

\[P\left(A^C \right) = 1 - P\left(A \right)\]

It is important to note that the superscript symbol used to denote “not” or “complement” differs from source to source.

Additive Rule

We can add the probabilities for two events together if they are disjoint or mutually exclusive. That is,

\[P \left( A \bigcup B \right) = P \left( A \right) + P \left( B \right)\]

Add probabilities for disjoint events.

but if the two events can occur simultaneously, then we know that \(P \left( A \bigcap B \right)>0\) and that we must then avoid double counting the intersection, making our full additive rule:

\[P \left( A \bigcup B \right) = P \left( A \right) + P \left( B \right) - P \left( A \bigcap B \right)\]

Don’t just add probabilities for events that are not mutually exclusive.

Multiplicative rule for independent events

This is often called conditional probability because we express the inter-relationship between events as a “condition”.

For events A and B where \(P(B)> 0\), \[P(A \vert B) = \frac{P \left( A \bigcap B \right)} {P \left( B \right)}\]

That is, the probability of A occuring, given B has also occurred, is equal to the probability that both A and B occur, divided by the probability that B occured.

Multiplicative rule for independent events

For independent events A and B, \(P(A \bigcap B) = P(A)P(B)\).

That is, we do not need to think about the conditional impact of either A or B on each other’s probability of occurrence when we multiply their separate probabilities together to find the joint probability of them both occurring.

Probabilities for events in series

Events in series are either independent or require use of conditional probability. Use the appropriate rule.

Probabilities for events in parallel

Events in parallel are almost always assumed to be independent of one another. In words we would be asking what the probability of at least one event happening would be, and can therefore use the additive rule given above for simple situations involving two parallel events.

Sometimes we are interested in more complicated situations though and we need to use the complementary rule and the multiplicative rule in conjunction. “At least one of…” is the complement of “none of…” so we can then use the right multiplicative rule from above.