Definitions

  • Disjoint (mutually exclusive) - Two events are mutually exclusive if they can't happen at the same time.
    • .
    • .
    • .
  • Exhaustive - A set of events is exhaustive if at least one event must occur.
    • .
    • .
  • Independent - Two events are independent if the occurrence of one doesn't affect the probability of the occurrence of the other.
    • .
    • - Knowing that happened doesn't change the probability of .
    • - Knowing that happened doesn't change the probability of .
  • Partition - A set of events form a partition if they're disjoint (mutually exclusive) and exhaustive exhaustive.

Combinatorics

Binomial Coefficients (1)

A binomial coefficient represents the number of ways you can choose an unordered subset of elements from a fixed set of elements. It is read as " choose ."

  • - The size of the total population.
  • - The number of objects being chosen from the total population.
Examples

If a sample is created by choosing a specific number of items from multiple groups within the population, the total number of ways to form that sample is the product of the binomials for each group:

  • - The sizes of the different groups.
  • - The number of objects chosen from each group.
Examples

Rule of Product

If one independent event can occur different ways, and another event can occur different ways, the number of combinations of both events occurring in sequence is:

Examples

The number of possible combinations of characters in a sequence characters long is:

  • - The number of available characters.
  • - The length of the character sequence.
Examples

Complement Rule

Examples

Conditional Probabilities

A conditional probability is the probability of an event occurring given that another event has already occurred. It is read as "the conditional probability of given ."

Examples

Bayes' Theorem (1)

Bayes' theorem is a formula for inverting a conditional probability:

Examples

Complements

Examples

Different Targets

Examples

Different Conditions

Examples

Multiplication Rule

You can find the probability of the intersection of events and by multiplying a conditional probability between them with the probability of the event which is the condition:

Independent Events

If two events and are independent, then a conditional probability between them will equal the probability of the target because knowing whether or not the condition has occurred doesn't change the probability:

Special Cases

The probability that exactly one of three events have occurred given that at least one of three events has occurred is described with this conditional probability:

De Morgan's Laws

Examples

Hypergeometric Distribution (1)

The hypergeometric distribution describes the probability of drawing an exact number of objects in a sample of size which have a certain characteristic that's present in a population of objects out of a total population of objects. The probability mass function for the hypergeometric distribution is:

  • - The size of the total population.
  • - The number of objects with a certain characteristic in the total population.
  • - The size of the sample being taken of the total population.
  • - The number of objects with the same characteristic in the sample.
Examples

Inclusion-Exclusion Principle

Disjoint (Mutually Exclusive)

Examples

Non-Disjoint (Non-Mutually Exclusive)

Examples

Examples

Examples

Examples

Independent Events

Complements

Examples

Mean

The mean for a data set is equal to the sum of all of the data points divided by the number of data points:

  • - The mean.
  • - The number of data points.
  • - Represents the position of a value in the dataset.
  • - A value in the dataset which is at position .
Examples

Sample Variance (Mean-Squared-Error)

The formula for the Mean-Squared-Error (MSE) is used to find the sample variance, which is equivalent to the standard deviation squared:

  • - The sample variance, which is equivalent to the standard deviation squared.
  • - The sample size.
  • - Represents the position of a value in the data set.
  • - A value in the dataset which is at position .
  • - The population mean.
Examples

Alternatively, if you've been given the summation of all the data points and the summation of all the data points squared you can use this formula to find the MSE:

Examples

Set Identities

Examples

One Complement

Two Complements

Total Probability

The law of total probability describes the probability of an event over a sample space that's partitioned into disjoint events. The formula states that the probability of can be found by summing all cases where could occur multiplied by the probability of each case:

Two Partitions and Two Events

This is what the formula would look like if there were two events, and , and the sample space was partitioned by and the complement :

This is what a graph of this situation would look like:

A

A'

B

Product

B'

Product

B

Product

B'

Product

P(A)

P(A')

P(B|A)

P(B'|A)

P(B|A')

P(B'|A')

‎

P(A∩B)

P(A∩B')

P(A'∩B)

P(A'∩B')

Examples