3.4 Probability Rules
The following are some basic rules of probability which can be easily verified with a Venn diagram:
- Complement Rule: [latex]P(\text{not }E) = 1 - P(E)[/latex] or [latex]P(E) = 1 - P(\text{not }E)[/latex].
- General Addition Rule: [latex]P(A \text{ or } B) = P(A) + P(B) - P(A \: \& \: B)[/latex].
- Special Addition Rule: when two events A and B are mutually exclusive, [latex]P(A \text{ or } B) = P(A) + P(B)[/latex].
This is due to the fact that [latex]P(A \: \& \: B) = 0[/latex], because it is impossible to observe both A and B.
More generally, if events [latex]A, B, C, \cdots[/latex] are mutually exclusive, then [latex]P(A \text{ or } B \text{ or } C \text{ or } \cdots) = P(A) + P(B) + P(C) + \cdots.[/latex]
Examples: Probability Rules
Suppose we roll a fair die, so that the sample space is S = {1, 2, 3, 4, 5, 6}. Consider the following events:
- Observing a six, E = {6}.
- Observing an even number, A = {2, 4, 6}.
- Observing an outcome that is less than 3, B = {1, 2}.
- Observing an odd number, C = {1, 3, 5}.
Since the die is fair, each outcome is equally likely. Therefore we can use the f/N rule to find the probabilities.
[latex]P(E) = \frac{f}{N} = \frac{1}{6}; \quad P(A) = \frac{f}{N} = \frac{3}{6},[/latex]
[latex]P(B) = \frac{f}{N} = \frac{2}{6}; \quad P(C) = \frac{f}{N} = \frac{3}{6}.[/latex]
Find the probabilities of the following events:
- (not E). Using the complement rule, [latex]P(\text{not }E) = 1 - P(E) = 1 - \frac{1}{6} = \frac{5}{6}[/latex].
- (A & C). Events A and C are mutually exclusive, since it is impossible to observe a number that is both even and odd. Therefore, [latex]A \mbox{ & } C=\varnothing[/latex] and [latex]P(A \: \& \: C) = 0[/latex].
- (A or B). Since events A and B are NOT mutually exclusive, we use the general addition rule. The overlap of events A and B is {2}, that is, A & B = {2} and therefore, [latex]P(A \: \& \: B) = \frac{f}{N} = \frac{1}{6}[/latex].
By the general addition rule,
[latex]P(A \text{ or }B) = P(A) + P(B) - P(A \: \& \: B) = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6}[/latex].
Since we are rolling a fair die, we can also use the [latex]\frac{f}{N}[/latex] rule to find the probabilities in the previous example.
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- (not E) = not observing a six={1, 2, 3, 4, 5}. [latex]P(\text{not }E)= \frac{f}{N} = \frac{5}{6}[/latex].
- (A or B) = observing an even number or a number not more than 3 = {1, 2, 4, 6} [latex]P(A \text{ or }B)= \frac{f}{N} = \frac{4}{6}[/latex].
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The results are identical to those using the probability rules.
