This section will provide the basic terms and properties associated with classical probability. We generally focus on classical probability but the probability properties apply to classical and subjective probabilities.
- Probability of an event
Probabilities will always be between (and including) 0 and 1. A probability of 0 means that the event is impossible. A probability of 1 means an event is guaranteed to happen. A probability close to 0 means the event is "not likely" and a probability close to 1 means the event is "highly likely" to occur. We denote the probability of event A as P(A).
- \(0 \leq P(A) \leq 1\)
- Range of probability for any event A.
- Probability of a complement
If A is an event, then the probability of A is equal to 1 minus the probability of the complement of A, $A^\prime$.
- We can see from the formula that \(1=P(A)+P(A^\prime)\).
- Probability of the empty set
If A and B are mutually exclusive, then $A\cap B=\emptyset$. Therefore, $P(A\cap B)=0$. This is important when we consider mutually exclusive (or disjoint) events.
- $P(A\cap B)=0$
- Probability of the union of two events
\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
If A and B are mutually exclusive, then \(P(AUB)=P(A)+P(B)\).
Try It! Probability Properties Section
Directions: Use the information given above to work out your answer to the questions below.
Given $P(A) = 0.6$, $P(B) = 0.5$, and $P(A\cap B)=0.2$.
Find \(P(A \cap B^\prime)\).
\(P(A \cap B^\prime)=P(A)-P(A\cap B)=0.6-0.2=0.4\)
Find \(P(B \cap A^\prime)\).
\(P(B \cap A^\prime)=P(B)-P(A\cap B)=0.5-0.2=0.3\)
Find \(P(A \cup B)\)\(P(A \cup B)=P(A)+P(B)-P(A \cap B)=0.6+0.5-0.2=0.9\)