18 3.1 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.
Example of an experiment: Flipping one fair coin twice.
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 S is used to denote the sample space.
Example: if you flip one fair coin, S = {H, T} where H = heads and T = tails are the outcomes.
An event is any combination of outcomes. Upper case letters like A and B represent events. For example, if the experiment is to flip one fair coin, event A might be getting at most one head.
Example: The probability of an event A is probability of getting at most one head. It is also written as P(A).
The probability of any outcome is the long-term relative frequency of that outcome. Probabilities are between zero and one, inclusive (that is, 0 [latex]\leq[/latex] probability of an event [latex]\leq[/latex] 1).
- P(A) = 0 means the event A can never happen.
- P(A) = 1 means the event A always happens.
- P(A) = 0.5 means the event A is equally likely to occur or not to occur.
Example: If you flip one fair coin repeatedly (from 20 to 2,000 to 20,000 times) the relative frequency of heads approaches 0.5 (the probability of heads).
Equally likely means that each outcome of an experiment occurs with equal probability.
Example:
- If you toss a fair, six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face.
- If you toss a fair coin, a Head (H) and a Tail (T) 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 A when all outcomes in the sample space are equally likely, count the number of outcomes for event A and divide by the total number of outcomes in the sample space.
For examples:
- If you toss a fair dime and a fair nickel, you will see four possible outcomes. These 4 outcomes will form a sample space. Therefore, the sample space is {HH, TH, HT, TT} where T = tails and H = heads.
If event A = getting one head, then there are two outcomes that meet this condition {HT, TH}.
The probability of event A, P(A) = [latex]\frac{\text{number of outcome with only one head}}{total of possible outcomes}[/latex] = [latex]\frac{2}{4}[/latex] = 0.5.
- Suppose you roll one fair six-sided die, with the numbers {1, 2, 3, 4, 5, 6} on its faces. Let event E = rolling a number that is at least five.There are two outcomes {5, 6}.
P(E) = [latex]\frac{\text{number of outcome that rolling a number that is at leas