# 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