6.1:
Probability in Statistics
Probability is the branch of mathematics that deals with the chances of an event occurring.
Consider the possible outcomes of tossing two quarters—head-head, head-tail, tail-head, or tail-tail.
Note that two out of four outcomes have one head and one tail.
In probability, each collection of outcomes is called an event, and those that cannot be broken into simpler components are called simple events.
The probability of an event is given by the number of ways it can occur divided by the total number of different simple events. It can be calculated for each case.
For any event, its probability can range between 0 and 1. For an impossible event, it is 0, and for a certain event, it is 1.
Probability is highly useful in statistics. Using laws of probability, statisticians can draw inferences from past events and predict future outcomes.
For instance, the computed probabilities of the coin toss experiment can be used to construct a probability distribution.
Comparing actual outcomes with these theoretical probabilities will determine if the outcomes are unusual.
6.1:
Probability in Statistics
Probability is the likelihood of an event occurring. The term event is defined as a collection of results of a procedure. An event is a simple event when an outcome cannot be divided into simpler parts.
An example of a simple event is a coin toss. The result of a coin toss is either a head or a tail. Here, head and tail are two simple events. These two simple events make up the sample space. Further, the probability of an event occurring falls within the range of 0 to 1. The probability of an impossible event is 0, whereas that of an event that would undoubtedly occur is 1.
If two coins are tossed, there are four likely outcomes. They are– head and head, head and tail, tail and head, and tail and tail. These four outcomes cannot be broken down further and are said to be simple events. Notice that two outcomes have one head and one tail. Only one outcome has either two heads or two tails – with this information, the probability can be calculated using the following equation:
In the equation, A is the event, s is the number of ways an event can occur, and n is the number of simple events.
In the coin toss experiment, the value of s for two heads is one; for two tails, it is one; and for a head and tail, it is two. The number of events, n, is 4. Using the equation, the probability of two heads in the coin toss is 1/4; two tails are 1/4, while that of a head and a tail is 2/4.
Furthermore, probability is a practical statistical tool. It can help statisticians predict future outcomes based on past events. A few of its applications lie in forecasting the weather, framing game, and sports strategies, and buying insurance.
This text is adapted from Openstax, Introductory Statistics, Section 3.1 Terminology under Probability Topics