Statistics, the art and science of learning from data by A. Agresti (fourth edition) – Chapter 5 summary

HOW PROBABILITY QUANTIFIES RANDOMNESS
Probability is the way we quantify uncertainness. It measures the chances of the possible outcomes for random phenomena. A random phenomenon is an everyday occurrence for which the outcome is uncertain. With random phenomena, the proportion of times that something happens is highly random and variable in the short run, but very predictable in the long run. The law of large numbers states that if the number of trials increases, the proportion of occurrences of any outcome approaches a given number. The probability of a particular outcome is the proportion of times that the outcome would occur in a long run of observations.

Different trials of a random phenomena are independent if the outcome of any one trial is not affected by the outcome of any other trial (e.g: if you have three children who are boys, the chance of the next child being a girl is not higher, but still ½).

In the subjective definition of probability, the probability is not based on objective data, but rather subjective information. The probability of an outcome is defined to be a personal probability. This is called Bayesian statistics.

FINDING PROBABILITIES
The sample space is the set of all possible outcomes (e.g: with being pregnant, the sample space is: {boy, girl}). An event is a subset of the sample space. An event corresponds to a particular outcome or a group of possible outcomes (e.g: a particular outcome or a group of possible outcomes). The probability of an event has the following formula:

For example, if you want to know the probability of the event throwing 6 with a fair dice, you calculate it like this:

• Number of outcomes in event A: 1 (there is only one possibility to throw 6)
• Number of outcomes in the sample space: 6 (you can throw between 1 and 6)
• P(A) = 1/6

The rest of the sample space for event A is called the complement of A. The complement of an event consists of all outcomes in the sample space that are not in the event.

Events that do not share any outcomes in common are disjoint (e.g: two events, A and B, are disjoint if they do not have any common outcomes). The chance that in the case of two events, A and B, both occur is called the intersection. The event that the outcome is A or B is the union of A and B.

There are three general rules for calculating the probabilities:

1. Complement rule
   
2. Addition rule
   There are two parts of the addition rule. For the union of two events:
   
   If the events are disjoint:
   
3. Multiplication rule (only works with independent trials)
   

CONDITIONAL PROBABILITY: THE PROBABILITY OF A GIVEN B
Conditional probability deals with finding the probability of an event when you know that the outcome was in some particular part of the sample space. It is most commonly used to find a probability about a category for one variable (e.g: a person being a drug user).

For events A and B, the conditional probability of event A, given that event B has occurred, is:

A multiplication rule in which it doesn’t matter if the events are independent is:

It is possible to calculate whether events are dependent or independent. An event is independent if:

APPLYING THE PROBABILITY RULES
A probability model specifies the possible outcomes for a sample space and provides assumptions on which the probability calculations for events composed of those outcomes are based. E PROBABILITY RULES

This table has some terminology attached to it:

Sensitivity and specificity refer to correct test results, while false positive rate and false negative rate refer to false test results. The prevalence is the base rate.