Lecture 10: Probability Theory Flashcards
What is Probability?
These are statements about outcomes which are possible but may not be guaranteed.
How can probabilities be expressed?
Through:
- fractions
- ratios
- percentages
Why is probability important?
- Allows us to reason about a computer sustems reliability
- What is the likelohood of a critical component failing?
- where should we build in redundancy?
- Helps describe a network server availability
List other examples were probability theory is used
- data mining
- pattern recognition
- encryption
- tracking
- robotics
- IS
- data compression
- Bayesian networks
- automatic classifciation
- information fusion
what are other factors of probability?
- Can be considered as “relative frequency”
- usually expressed as real numbers ranging between zero and one
what does the term experiement mean?
It is a process which generates data
- a single performance of the experiment is call atrial
- each trial has an outcome
- e.g. tossing a coin
What does the term sample space mean?
The set of all possible outcomes
- e.g single coin tose (h, t)
- e.g2. coin tossed twice (hh, ht etc…)
What does the term event mean?
A subset of the sample space
- the outcome with particular characteristics
- e.g. getting the same result in two coin tosses (HH,TT)
Describe discrete probability.
considers only experiments for which the sample space contains a finite number of elements or a countably infinite number of elements.
What is a probability density function?
assigns probabilities to all of the elements in the sample space
Explain Uniform probabilities. (uniform probability density functions)
If a sample space contains N elements, all of which are likely to occur, then probability assigned to each are 1/N.
What does it mean by “Counting sample points”?
some problems need to be solved by specifically counting the number of elements in a set or subset, without actually listing each element
list the 7 basic counting strategies.
- addition principle
- subtraction principle
- pigeon-hole principle
- complementation
- combinations
- permutations of distinct objects
- permutations of repeated objects
Addition.
suppose one element is to be selected from a collection of disjoint or mutually exclusive sets A1, A2…An
then the total amount of possible choices is given by A1 + A2 + An…
What does it mean to be mutually exclusive set?
to have no elements in common
Multiplication
a process can be performed in a sequence of m distinct steps.
total outcomes: n = n1 x n2…..x Nm
permutations of distinct objects
When the order DOES MATTER in a counting problem
a permutation is a selection (without replacement), of items from a set, where the order of selection is important.
What is a circular permutation of distinct objects
- where items are aranged in a circle.
- There are no ends to the arrangements
- 2 arrangements can be similar if one can be transformed into another by a clockwise rotation of elements
- e.g. abc = cba (same circular permutation)
Combination
order DOES NOT MATTER
combination is an unordered selection of items in a set
Complementation
it may be easier calculation the number of sample points which do not have a particular characteristic
Complementary sets - a set which have a required characteristic and a set which dosent.
Pigeon-hole principle
- If m objects are to be placed in n locations and m > n > 0, then at least one location must receive two objects.
- IN counting context: if the total number of elements in a set, is larger then the total number of distinct properties, then at least 2 of the elements have the same property.
- e.g 15000 customers, 1000 PINS, 2 have the same
Joint probabilities
two or more events may include the same elements of the sample space
e.g. the characteristics defining the two events may be found in the sample space
Conditional Probabilities
Involves calculating the probability of an event when it is known that some other event has occurred.
e.g. A1 occurrence may depend on the occurrence of event A2
Independent Events
two events are independent if the probability of one, event is not affected by the occurrence of the other event
Binomial Distribution
occurs when
- repeated trials are independent
- probability of success is constant then, experiment is binomal
some experiments consist of repeated trials, where each trial has only two possible outcomes:
- heads or tails
- 0 or 1
- working or defective
- success or failure
Summarise. 5 main points to remember.
- Probability can be considered as the relative frequency of occurrence of an event
- The probability of an event is a value between 0 to
- The probabilities for all of the points in the sample space sum to 1
- To calculate the probability of an event you may need to:
- Count sample points
- Decide if events are conditional or independent
- The binomial distribution is useful for calculating probabilities of events associated with a particular type of experiment
