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Chapter 13
COMPSCI 70 Chapter Notes  Chapter 13: Probability Space, List Of Poker Hand Categories, Coin Flipping
by OC1339993
Department
Computer ScienceCourse Code
COMPSCI 70Professor
Rao SatishChapter
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Probability is a ratio of counting
○
The "numerators" and "denominators" or probabilities are calculated by counting

Probabilities are restricted to rational numbers

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
1)
event space  a collection of events (subsets of the sample space)
2)
The probability of each sample point in a sample space is
for all
1)
All sample points are countable
(the sum of the probabilities of all outcomes/sample points is 1)
2)
Conditions of a probability function
probability function  a function that assigns a probability to each outcome; measures the likelihood of each outcome
3)
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

If the coin has bias (for Heads), and we flip the coin times with exactly Heads, then the probability of the sequence is

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

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

The probability of any event is

Rolling dice
permutations of the deck

The probability space of random shuffling a deck of cards is uniform

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
○

The probability space is uniform

Event that the poker hand is a flush (all cards have the same suit)
Examples of poker hands probabilities

Poker hands
13. Intro to Discrete Probability
Thursday, October 5, 2017
6:42 AM
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