9: Probability
- Page ID
- 201177
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Mathematics is often seen as finding exact solutions to precise problems, so it may seem an unlikely tool for studying random events. But there is a well-developed and powerful branch of mathematics, called probability theory, that does exactly that. This chapter introduces you to some fundamental probability concepts and applications.
- 9.1: Sample Spaces and Probability
- Probability measures the likelihood of something happening. This section defines exactly how we can measure "likelihood" mathematically, and presents a number of examples of doing so.
- 9.2: Mutually Exclusive Events and the Addition Rule
- Because events are sets, we can construct them using the union, intersection, and complement set operations. The probabilities of such "compound" events are related to the probabilities of the events from which we constructed them, but not always in the most obvious ways. This section begins the study of compound probabilities by examining unions and complements of events.
- 9.3: Probability Using Tree Diagrams and Combinations
- Finding the number of outcomes in an event, and the total number in a sample space, involves counting. Thus probability problems often have counting problems hidden within them, and the counting techniques we developed earlier in this book become handy. This section explores applications of such counting techniques as tree diagrams and combinations to probability.
- 9.4: Conditional Probability
- The probability of an event sometimes depends on whether another event also happens. So-called "conditional probability" allows us to describe and calculate probabilities in the presence of such relationships. This section defines conditional probability, related notations, and formulas for calculating it.
- 9.5: Independent Events
- Intuitively, saying that two events are "independent" should mean that neither affects the probability of the other. This section gives a mathematical meaning to this intuition and explores its consequences.
- 9.6: Bayes' Formula
- Conditional probability is helpful in real-world settings where we observe some event, A, happening, and want to know the probability of event B following. Surprisingly often though, we want to reverse this order: we observe event B, and want to know the probability that A preceded it. This section introduces Bayes' Formula, a handy tool for solving exactly this sort of problem.
- 9.7: Binomial Probability
- A series of simple experiments can produce rich patterns in the series of outcomes, even if the choices for each individual outcome are very limited. Calculating the probability that a pattern will occur in a series of outcomes is similarly harder than calculating the probability of a single outcome. The calculations are nonetheless often doable, as seen in this section.
- 9.8: Expected Value
- The average of a set of numbers is their sum divided by the number of numbers. But this average loses much of its practical significance if there are probabilities attached to the numbers, so that they are not all equally likely to appear. In such situations, a statistic called "expected value" may be more useful than the simple average. This section defines what expected value is and how to find it.