Probability¶
Random Event¶
A random event is an event that can have different outcomes. E.g. when randomly selecting a penguin from a larger group, the species of the chosen penguin is the outcome.
Random events are often denoted by capital letters:
or by explicit labels:
The sum of the probabilities for all possible outcomes of a random event has to be 1.0.
Conditional Probability¶
Often, the probability of an outcome depends on the outcome of a different random event. This is called a conditional probability.
For instance, if the probability of outcome A depends on outcome B, you can write:
Prior¶
In contrast to a conditional probability, the probability of an event without any additional information is called prior probability or prior.
Total Probability¶
To calculate the prior probability \(p(A)\) from a conditional probability \(p(A|B)\), you have to sum up the conditional probabilities for all possible outcomes of B. This is called the total probability:
Multiplication Rule¶
If two events are consecutive and you want the probability of a combination, you need to multiply the probabilities with each other:
The \(\cap\) symbol denotes the intersection of both outcomes.
Bayes Theorem¶
With Bayes Theorem, you can reverse the order of a conditional probability:
Here, \(p(A)\) is the prior probability (if we had no additional information about A).
Statistical Independence¶
Two events are called independent if their conditional probabilities are equal to their respective priors:
for all \(B_i\). From Bayes Theorem it follows that in this case the reverse is also true: