# Probability

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- Sample spaces
- Collection of all possible outcomes, such as all six faces of a die or all 52 cards in a deck

- Events
- A simple event is an outcome from a sample space with one characteristic, such as a red card from a deck of cards
- A joint event involves two outcomes simultaneously, such as an ace that is also red from a deck of cards
- An impossible event is impossible, such as a black and red card, also known as a null event
- A complement of an event
- Mutually exclusive events cannot occur together
- Collectively exhaustive events means that one of the events must occur, such that the set of events covers the whole sample space

- Visualising events
- Contingency tables

Ace | Not an ace | Total | |

Black | 2 | 24 | 26 |

Red | 2 | 24 | 26 |

Total | 4 | 48 | 52 |

The sample space in this case is 52.

- Simple probability
- Probability is the numerical measure (between 0 and 1) of the likelihood that an event will occur. The sum of the probabilities of all mutually exclusive and collective exhaustive events is 1.
- The probability of an event E is calculated by -> P(E) = number of event outcomes/total number of possible outcomes
- Where each of the outcomes in the sample space is equally likely to occur

- Joint probability
- The probability of a joint event, A and B -> number of outcomes from both A and B/total number of possible outcomes

Event | B_1 | B_2 | Total |

A_1 | P(A_1 and B_1) | P(A_1 and B_2) | P(A_1) |

A_2 | P(A_2 and B_1) | P(A_2 and B_2) | P(A_2) |

Total | P(B_1) | P(B_2) | 1 |

- Compound probability
- Probability of a compound event, A or B -> number of outcomes from either A or B or both/total number of outcomes in sample space
- P(A_1 or B_1) = P(A_1) + P(B_1) - P(A_1 and B_1)
- For mutually exclusive events, P(A or B) = P(A) + P(B)

- Conditional probability
- The probability of event A given that event B has occurred
- P(A|B) = P(A and B)/P(B)
- Multiplication rule -> P(A and B) = P(A|B) . P(B) = P(B|A) . P(A)

- Statistical independence
- Events A and B are independent when the probability of one event, A, is not affected by another event, B
- Events A and B are independent if P(A|B) = P(A) or P(B|A) = P(B) or P(A and B) = P(A) . P(B)

- Random variable
- Random variables are outcomes of an experiment expressed numerically
- A discrete random variable is obtained by counting and usually has a finite number of different values

- Discrete probability distribution
- List of all possible [X_j, P(X_j)] pairs, where X_j = value of random variable and P(X_j) = probability associated with random variable
- They are mutually exclusive, meaning nothing in common
- They are collectively exhaustive, meaning nothing is left out
- 0 <= P(X_j) <= 1 and the sum of P(X_j) = 1