Statistics for Psyc PDF

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Psyc notes for stats probability ...

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Probability is a concept which quantifies the chance of a future event occurring.  e.g. the chance that I get a head when I flip a coin  the chance that it rains this evening  the chance that a new business survives for at least two years  The notation used is P(A) = “the probability that A occurs” 5.1 Interpretation 

The most useful interpretation is that if we repeat our ‘experiment’ many many times: probability of an outcome  = long run relative frequency of outcome  = no of times outcome occurs total no of trials  “Long run” implies no. trials is large.  Fine when tossing coins, but not always useful, e.g. P(rain tomorrow) will be a subjective (or personal) probability - we cannot repeat this “experiment”.  e.g. coin tossing experiment 5.1.2 Certain and impossible events   

A certain event has probability one, or 100%. An impossible event has probability zero, or 0%. We should distinguish between numbers close to these extremes, and the extremes themselves. Very unlikely events can still happen.  All other events have probabilities between 0% and 100%, i.e. 0 ≤ P(A) ≤ 1  for any event A. P(A) can be written as a decimal, fraction, or percentage 5.1.3 Outcomes and events  

Probabilities are defined for outcomes or events. Outcomes are the things that can happen in an experiment at its most fundamental level, and are listed in an outcome space - a set usually labelled Ω.  e.g. tossing a coin, Ω = {H, T}  e.g. rolling a die, Ω = {1, 2, 3, 4, 5, 6}  Events are collections of outcomes.  e.g. A = {1, 3, 5}  The event occurs if any one of the outcomes in it occurs.  e.g. A = { odd number when roll a die } 5.1.4 Equally likely outcomes 

When the outcomes are equally likely (as in tossing a fair coin or rolling a fair die), the probability of any outcome is  P(outcome) = 1/number of outcomes  Similarly, the probability of any event based on these outcomes is P(event) = number of outcomes in event/ number of outcomes  e.g. Consider rolling two (fair) dice. What is the outcome space? What is the probability of any outcome? What is the probability that the sum on the dice equals seven? 5.1.5 Combining events   

There are two common combinations of events: union, which is the collection of outcomes in either event. Commonly written A∪B, and said “A or B”. The union occurs when any outcome in A or B (or both) occurs. intersection, which is the collection of outcomes in both events. Commonly written A P (Pacific ∪ Other) B, and said “A and B”. The intersection occurs if any outcome in both A and B occurs.

5.1.6 Venn diagrams   

Venn diagrams are useful ways of graphically representing events and their interrelationships. The outcome space Ω is represented by a large rectangle. Events are drawn as circles. which usually overlap to represent intersections between these event e.g. Draw a Venn diagram with two events A and B, and identify all regions in the outcome space Ω.

5.1.7 Mutually exclusive events  

Two events which have no outcomes in common are said to be mutually exclusive. The intersection of two mutually exclusive events is the empty set, and in a Venn diagram, the events can be shown as nonoverlapping.  The probability that both mutually exclusive events happens is zero, i.e. it is impossible  The outcomes of any experiment are themselves mutually exclusive, since no two outcomes can occur at once.  e.g. Suppose you have a group of 100 individuals. 63 of them own a car, and 24 of them own a bicycle. You select an individual at random. What are the following probabilities? 5.1.8 Adding probabilities 

If events A and B are mutually exclusive, the probability that A or B occurs is the sum of P(A) and P(B), i.e P(A ∪ B) = P(A) + P(B) (provided P(A ∩ B) = 0). In general, i.e. when we don’t know that they are mutually exclusive P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

    5.1.9 Complementary events    

The complement of any event is the set of all outcomes not in the event. We define A0 to be the complement of A, or “not A”. Since A ∪ A 0 = Ω i.e. one or other must happen, and they’re mutually exclusive (i.e. can’t both happen) i.e. one or other must happen, and they’re mutually exclusive (i.e. can’t both happen) P(A) + P(A 0 ) = 1 or P(A 0 ) = 1 − P(A)

5.1.10 Conditional probability   

If we know event B has occurred, the probability that A also occurs is the conditional probability of A given B. This is typically written P(A given B) or P(A | B), and the general formula is P(A given B) = P(A and B)/P(B) P(A and B) is rescaled to reflect that B must have happened. Note that P(B) cannot equal zero, mathematically or intuitively

5.1.11 Independent events 

If two events are independent, the outcome of one has no bearing on the probability of the other. e.g. the numbers shown on two dice rolled together.

 Thus P(A given B) = P(A) and P(B given A) = P(B)  If these two things are true, then it is also true that P(A and B) = P(A) × P(B)  for independent events A and B. 5.1.12 Multiplying probabilities   

Whether conditional (dependent) or independent, we multiply probabilities to find the probability that both events occur, i.e. P(A and B) = P(A) × P(B given A) in general, or P(A and B) = P(A) × P(B) in the special case that A and B are independent. NB: When multiplying, probabilities should be decimal or fraction, but not percentage.

5.1.13 Probability trees 

A sequence of events, as in the last example, can be very conveniently represented using a probability tree. These can be drawn for conditional or independent events.

5.1.14 Features of the probability tree  

From a starting point, the probability tree shows: A pathway of branches to all possible outcomes • each branch is labelled with its (conditional) probability  The probability of a particular outcome is found by multiplying the probabilities along the branches to that outcome  The probability of events (combinations of outcomes) can be found by adding the probabilities of each outcome in that event. 5.1.15 Bayes’ rule 

 

Often we are interested in the probability of observing the first event, given a particular occurrence of the second event - the opposite order that the information is provided. Bayes’ rule tells us how to find this probability. P(A given B) = P(A and B)/P(B) We will typically use a tree diagram to find the probabilities P(A and B) and P(B)....