alt="p equals 1 slash 2"/>, we say the coin is fair. If In a mathematical sense, all of these coin tossing models are “correct” in that they are consistent with the definition of what a probability is. However, we might debate which model most accurately reflects reality and which is most useful for modeling actual coin tosses.
Example 1.6 Suppose that a college has six majors: biology, geology, physics, dance, art, and music. The percentage of students taking these majors are 20, 20, 5, 10, 10, and 35, respectively, with double majors not allowed. Choose a random student. What is the probability they are a science major?The random experiment is choosing a student. The sample space isThe probability model is given in Table 1.1. The event in question isFinally,
TABLE 1.1. Probability model for majors.
| Bio | Geo | Phy | Dan | Art | Mus |
|---|---|---|---|---|---|
| 0.20 | 0.20 | 0.05 | 0.10 | 0.10 | 0.35 |
This example is probably fairly clear and may seem like a lot of work for a simple result. However, when starting out, it is good preparation for the more complicated problems to come to clearly identify the sample space, event, and probability model before actually computing the final probability.
Example 1.7 In three coin tosses, what is the probability of getting at least two tails?Although the probability model here is not explicitly stated, the simplest and most intuitive model for fair coin tosses is that every outcome is equally likely. As the sample spacehas eight outcomes, the model assigns to each outcome the probability The event of getting at least two tails can be written as This gives
1.4 PROPERTIES OF PROBABILITIES
Events can be combined together to create new events using the connectives “or,” “and,” and “not.” These correspond to the set operations union, intersection, and complement.
For sets
In probability word problems, descriptive phrases are typically used rather than set notation. See Table 1.2 for some equivalences.
A Venn diagram is a useful tool for working with events and subsets. A rectangular box denotes the sample space
One of the most basic, and important, properties of a probability function is the simple addition rule for mutually exclusive events. We say that two events are mutually exclusive, or disjoint, if they have no outcomes in common. That is,
TABLE 1.2. Events and sets.
| Description | Set notation |
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Either |
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Not |
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