4: Basic Concepts of Probability: The math
- Page ID
NOTE: The following chapter is from an Introductory Statistics book by Shafer and Zhang. You may notice a slight change of tone/voice in the way it is written.
Suppose a polling organization questions 1,200 voters in order to estimate the proportion of all voters who favor a particular bond issue. We would expect the proportion of the 1,200 voters in the survey who are in favor to be close to the proportion of all voters who are in favor, but this need not be true. There is a degree of randomness associated with the survey result. If the survey result is highly likely to be close to the true proportion, then we have confidence in the survey result. If it is not particularly likely to be close to the population proportion, then we would perhaps not take the survey result too seriously. The likelihood that the survey proportion is close to the population proportion determines our confidence in the survey result. For that reason, we would like to be able to compute that likelihood. The task of computing it belongs to the realm of probability, which we study in this chapter.
- 4.1: Sample Spaces, Events, and Their Probabilities
- The sample space of a random experiment is the collection of all possible outcomes. An event associated with a random experiment is a subset of the sample space. The probability of any outcome is a number between 0 and 1. The probabilities of all the outcomes add up to 1. The probability of any event A is the sum of the probabilities of the outcomes in A.
- 4.2: Complements, Intersections, and Unions
- Some events can be naturally expressed in terms of other, sometimes simpler, events.
- 4.3: Conditional Probability and Independent Events
- A conditional probability is the probability that an event has occurred, taking into account additional information about the result of the experiment. A conditional probability can always be computed using the formula in the definition. Sometimes it can be computed by discarding part of the sample space. Two events A and B are independent if the probability P(A∩B) of their intersection A ∩ B is equal to the product P(A)⋅P(B) of their individual probabilities.
- 4.E: Basic Concepts of Probability (Exercises)
- These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang.