Probability Concepts
Trials and Outcomes
In probability theory, a trial refers to a single execution of an experiment or a random process. Each trial results in an outcome, which is a specific result of that trial.
ExampleFlipping a coin is a trial, and the outcome could be either heads or tails. Rolling a die is another trial, with possible outcomes being any number from 1 to 6.
Equally Likely Outcomes
Outcomes are considered equally likely if they have the same probability of occurring. This concept is fundamental in many probability calculations.
NoteIn a fair die, each number has an equal 1/6 probability of being rolled, making all outcomes equally likely.
Relative Frequency
Relative frequency is the ratio of the number of times an event occurs to the total number of trials conducted. It's often used as an experimental approximation of probability.
$$ \text{Relative Frequency} = \frac{\text{Number of occurrences of the event}}{\text{Total number of trials}} $$
ExampleIf you flip a coin 100 times and get 53 heads, the relative frequency of heads is 53/100 = 0.53.
Sample Space (U)
The sample space, denoted as U, is the set of all possible outcomes of an experiment.
ExampleFor a single coin flip: U = {Heads, Tails} For rolling a six-sided die: U = {1, 2, 3, 4, 5, 6}
Sample spaces can be represented in various ways:
- Lists: U = {1, 2, 3, 4, 5, 6}
- Tables:
- Tree diagrams:
Representing sample spaces visually can help in understanding complex probability scenarios.
Events
An event is a subset of the sample space. It's a collection of outcomes that we're interested in.
