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Chapter 13

# COMPSCI 70 Chapter Notes - Chapter 13: Probability Space, List Of Poker Hand Categories, Coin Flipping

Department
Computer Science
Course Code
COMPSCI 70
Professor
Rao Satish
Chapter
13

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Probability is a ratio of counting
The "numerators" and "denominators" or probabilities are calculated by counting
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Probabilities are restricted to rational numbers
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probability - likelihood; the use of models to reason about uncertainty
discrete probability - involves outcomes that are all discrete values (
 ) (all sample points are countable); the probability of any event is completely determined once the probability of each outcome is specified
sample space
event space - a collection of events (subsets of the sample space)
The probability of each sample point in a sample space is   
 
  
 for all   
All sample points are countable

 
 (the sum of the probabilities of all outcomes/sample points is 1)
Conditions of a probability function
probability function - a function that assigns a probability to each outcome; measures the likelihood of each outcome
Three components of a probability space
For uniform spaces, computing probabilities reduces to counting the number of sample points
uniform proability space - a probability space in which all its sample points have the same probability, which is
If
 , then 
  
The probability of a sample point is one fraction of the total number of sample points
For each of the sample points/outcomes in the space, each can be either included or excluded from an event.
The maximum number of events in an uniform probability space where
  is
Ex. flipping a fair coin, dealing a poker hand that is uniform
Uniform distribution of probability space
probability space - the space that contains a sample space, event space, and probability function of a random experiment; constructed with a specific kind of situation or experiment in mind; models uncertainty
The sample space and the probabilities specify the random experiment
The random experiment selects/results in one and only one outcome in the sample space
The set of all possible combinations of the elements in A and the elements in B
 

    
 
Notation
sample space - the space/set of all outcomes (sample points); the complete experiement and its set of possible outcomes
Each sample point has a probability, 
Each sample point describes 1 outcome of the complete experiement
sample point - one outcome of a random experiment; one outcome of the sample space
If the sample space is uniform, then the probability of event A is
, where is the sample space
probability of an event, -

 (the sum of all probabilities of the sample points in A)
Each event is a set containing 0+ outcomes/sample points
The probabiltiy of both event A and B hapenning is P(A)P(B)
The two events do not affect the proability of the other
Events are independent if

 

Independent events
"if event occurs, then event occurs" means thaht event is a subset of event (i.e.   )
event -a subset of a sample space,   
 
as   
The average of random variable , denoted as
, eventually (as   ) converges to the expected value of just one of the values that takes on (just one of the values in 's image)
As the number of experiments increases, the actual ratio of outcomes will converge on the expected ratio of outcomes
Law of Large Numbers - a principle of probability according to the frequencies of events with the same likelihood of occurrence even out
CLT gives us the ability to approximate the distribution of certain statistics, even if we know very little about the underly ing sampling distribution
Central Limit Theorem: given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population
 

Stirling formula (for a large ): formula that gives the factorial of a large number
Probability theorems and laws
The cardinality of the sample space for flipping a coin times is
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If the coin has bias (for Heads), and we flip the coin times with exactly Heads, then the probability of the sequence is

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Event consists of exactly
sample points
Each sample point has the probability

If the event where we get exactly Heads when we flip the coin times, then

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Coin flipping
Each sample point in the sample space of rolling 2 die is

Possible outcomes of rolling 2 die:


    
The probability space of rolling two die is uniform
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The probability of any event is 


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Rolling dice
  permutations of the deck
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The probability space of random shuffling a deck of cards is uniform
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Card shuffling
The sample space is the set of all possible poker hands, which corresponds to choosing   objects without replacement from a size of size   

  
 
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The probability space is uniform
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Event that the poker hand is a flush (all cards have the same suit)
Examples of poker hands probabilities
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Poker hands
13. Intro to Discrete Probability
Thursday, October 5, 2017
6:42 AM
CS70 Page 1