Summary
Highlights
The sample space, usually denoted by Omega, represents the set of all possible outcomes of a random experiment. Using a single die as an example, the sample space is {1, 2, 3, 4, 5, 6}. For two dice, there are 36 possible outcomes, where the order of the dice matters (e.g., (1,2) is different from (2,1)).
An outcome is simply the result of a random experiment, like rolling a 6 or rolling a (1,3). An event, however, is a subset of the sample space, often a collection of several outcomes. For example, rolling an odd number on a single die is an event {1, 3, 5}.
Further examples of events include 'the number rolled is greater than 3' ({4, 5, 6}) and 'the number rolled is a prime number' ({2, 3, 5}). An event can also consist of a single outcome, or it can be an empty set if no outcome satisfies the condition (e.g., 'rolled number is greater than 8').
The complementary event (E-bar) consists of all outcomes not in the original event E. For instance, if event E is 'odd number' ({1, 3, 5}), its complementary event E-bar is 'even number' ({2, 4, 6}). It's crucial to correctly define the complement (e.g., the complement of 'greater than 3' is 'at most 3', not 'less than 3').
Probability is an estimation of how likely an event is to occur. It is always a number between 0 and 1. A probability of 0 means an event is impossible, while 1 means it's certain. Probabilities are often expressed as percentages by multiplying by 100% (e.g., 0.5 is 50%, 0.0236 is 2.36%).
Absolute frequency is the actual count of how many times something occurs. Relative frequency is the absolute frequency divided by the total number of trials or observations. For example, in a class of 10 boys and 15 girls (25 total), the absolute frequency for boys is 10, and the relative frequency is 10/25.
The empirical law of large numbers states that the more times an experiment is repeated, the closer the relative frequency of an event will get to its actual probability. For instance, when repeatedly throwing a Lego brick to see how it lands, the relative frequency of a particular landing position will approach its true probability as the number of throws increases.
Stochastics is the umbrella term for probability theory and statistics. This section introduces the first basic concepts, starting with the sample space.