# 2.4 - Probability Properties

2.4 - Probability Properties

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, $A^\prime$.

$$P(A)=1-P(A^\prime)$$
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

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 $P(A\cap B)=0.2$.

1. Find $$P(A^\prime)$$.
$$P(A^\prime)=1-P(A)=0.4$$
2. Find $$P(A \cap B^\prime)$$.

$$P(A \cap B^\prime)=P(A)-P(A\cap B)=0.6-0.2=0.4$$

3. Find $$P(B \cap A^\prime)$$.

$$P(B \cap A^\prime)=P(B)-P(A\cap B)=0.5-0.2=0.3$$

4. Find $$P(A \cup B)$$
$$P(A \cup B)=P(A)+P(B)-P(A \cap B)=0.6+0.5-0.2=0.9$$

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