The Terminology of Probability
Learning Outcomes
- Understand and use the terminology of probability
Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity. An experiment is a planned operation carried out under controlled conditions. If the result is not predetermined, then the experiment is said to be a chance experiment. Flipping one fair coin twice is an example of an experiment.
A result of an experiment is called an outcome. The sample space of an experiment is the set of all possible outcomes. Three ways to represent a sample space are: to list the possible outcomes, to create a tree diagram, or to create a Venn diagram. The uppercase letter [latex]S[/latex] is used to denote the sample space. For example, if you flip one fair coin, [latex]S[/latex] = {[latex]H[/latex], [latex]T[/latex]} where [latex]H[/latex] = heads and [latex]T[/latex] = tails are the outcomes.
An event is any combination of outcomes. Upper case letters like [latex]A[/latex] and [latex]B[/latex] represent events. For example, if the experiment is to flip one fair coin, event [latex]A[/latex] might be getting at most one head. The probability of an event [latex]A[/latex] is written [latex]P[/latex]([latex]A[/latex]).
The probability of any outcome is the long-term relative frequency of that outcome. Probabilities are between zero and one, inclusive (that is, zero and one and all numbers between these values). [latex]P[/latex]([latex]A[/latex]) = [latex]0[/latex] means the event [latex]A[/latex] can never happen. [latex]P[/latex]([latex]A[/latex]) = [latex]1[/latex] means the event [latex]A[/latex] always happens. [latex]P[/latex]([latex]A[/latex]) = [latex]0.5[/latex] means the event [latex]A[/latex] is equally likely to occur or not to occur. For example, if you flip one fair coin repeatedly (from [latex]20[/latex] to [latex]2,000[/latex] to [latex]20,000[/latex] times) the relative frequency of heads approaches [latex]0.5[/latex] (the probability of heads).
Equally likely means that each outcome of an experiment occurs with equal probability. For example, if you toss a fair, six-sided die, each face ([latex]1, 2, 3, 4, 5, \text{or}\,6[/latex]) is as likely to occur as any other face. If you toss a fair coin, a Head ([latex]H[/latex]) and a Tail ([latex]T[/latex]) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.
To calculate the probability of an event [latex]A[/latex] when all outcomes in the sample space are equally likely, count the number of outcomes for event [latex]A[/latex] and divide by the total number of outcomes in the sample space. For example, if you toss a fair dime and a fair nickel, the sample space is {[latex]HH[/latex], [latex]TH[/latex], [latex]HT[/latex], [latex]TT[/latex]} where [latex]T[/latex] = tails and [latex]H[/latex] = heads. The sample space has four outcom