Probability Rules!. ● Probability relates short-term results to long-term results ● An example ...
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Transcript of Probability Rules!. ● Probability relates short-term results to long-term results ● An example ...
![Page 1: Probability Rules!. ● Probability relates short-term results to long-term results ● An example A short term result – what is the chance of getting a.](https://reader035.fdocuments.us/reader035/viewer/2022062517/56649e7b5503460f94b7bc9e/html5/thumbnails/1.jpg)
5.1
Probability Rules!
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PROBABILITY
● Probability relates short-term results to long-term results
● An example A short term result – what is the chance of getting
a proportion of 2/3 heads when flipping a coin 3 times
A long term result – what is the long-term proportion of heads after a great many flips
A “fair” coin would yield heads 1/2 of the time – we would like to use this theory in modeling
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LONG TERM PROBABILITY
● Relation between long-term and theory The long term proportion of heads after a great
many flips is 1/2 This is called the Law of Large Numbers
● Relation between short-term and theory We can compute probabilities such as the chance
of getting a proportion of 2/3 heads when flipping a coin 3 times by using the theory
This is the probability that we will study
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DEFINITIONS
● Some definitions An experiment is a repeatable process where the
results are uncertain An outcome is one specific possible result The set of all possible outcomes is the sample
space● Example
Experiment … roll a fair 6 sided die One of the outcomes … roll a “4” The sample space … roll a “1” or “2” or “3” or “4”
or “5” or “6”
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DEFINITIONS CONTINUED
● More definitions An event is a collection of possible outcomes …
we will use capital letters such as E for events Outcomes are also sometimes called simple
events … we will use lower case letters such as e for outcomes / simple events
● Example (continued) One of the events … E = {roll an even number} E consists of the outcomes e2 = “roll a 2”, e4 =
“roll a 4”, and e6 = “roll a 6” … we’ll write that as {2, 4, 6}
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DIE ROLLING
Summary of the example The experiment is rolling a die There are 6 possible outcomes, e1 =
“rolling a 1” which we’ll write as just {1}, e2 = “rolling a 2” or {2}, …
The sample space is the collection of those 6 outcomes {1, 2, 3, 4, 5, 6}
One event is E = “rolling an even number” is {2, 4, 6}
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PROBABILITY…BETWEEN 0 AND 1
Rule – the probability of any event must be greater than or equal to 0 and less than or equal to 1 It does not make sense to say that there is a –
30% chance of rain It does not make sense to say that there is a
140% chance of rain Note – probabilities can be written as
decimals (0, 0.3, 1.0), or as percents (0%, 30%, 100%), or as fractions (3/10)
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SUM OF PROBABILITIES
Rule – the sum of the probabilities of all the outcomes must equal 1 If we examine all possible cases, one of
them must happen It does not make sense to say that there
are two possibilities, one occurring with probability 20% and the other with probability 50% (where did the other 30% go?)
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SPECIAL EVENTS
Probability models must satisfy both of these rules
There are some special types of events If an event is impossible, then its
probability must be equal to 0 (i.e. it can never happen)
If an event is a certainty, then its probability must be equal to 1 (i.e. it always happens)
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THAT’S UNUSUAL?!
● A more sophisticated concept An unusual event is one that has a low probability
of occurring This is not precise … how low is “low?
● Typically, probabilities of 5% or less are considered low … events with probabilities of 5% or lower are considered unusual
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PROBABILITY
● If we do not know the probability of a certain event E, we can conduct a series of experiments to approximate it by
● This becomes a good approximation for P(E) if we have a large number of trials (the law of large numbers)
experimenttheoftrialsofnumberoffrequency
)(E
EP
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PROBABILITY
Example We wish to determine what proportion of
students at a certain school have type A blood We perform an experiment (a simple random
sample!) with 100 students (this is empirical probability since we are collecting our own data)
If 29 of those students have type A blood, then we would estimate that the proportion of students at this school with type A blood is 29%
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EXAMPLE (CONTINUED)
We wish to determine what proportion of students at a certain school have type AB blood We perform an experiment (a simple
random sample!) with 100 students If 3 of those students have type AB blood,
then we would estimate that the proportion of students at this school with type AB blood is 3%
This would be an unusual event
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EQUALLY LIKELY OUTCOMES
● The classical method applies to situations where all possible outcomes have the same probability
● This is also called equally likely outcomes● Examples
Flipping a fair coin … two outcomes (heads and tails) … both equally likely
Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and 6) … all equally likely
Choosing one student out of 250 in a simple random sample … 250 outcomes … all equally likely
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EQUALLY LIKELY OUTCOMES
● Because all the outcomes are equally likely, then each outcome occurs with probability 1/n where n is the number of outcomes
● Examples Flipping a fair coin … two outcomes (heads and
tails) … each occurs with probability 1/2 Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and
6) … each occurs with probability 1/6 Choosing one student out of 250 in a simple
random sample … 250 outcomes … each occurs with probability 1/250
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CLASSICAL PROBABILITY
● What is “theoretically supposed to happen”● The general formula is
● If we have an experiment where There are n equally likely outcomes (i.e. N(S) = n) The event E consists of m of them (i.e. N(E) = m)
n
mEP )(
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CLASSICAL PROBABILITY
● Because we need to compute the “m” or the N(E), classical methods are essentially methods of counting
● These methods can be very complex!● An easy example first● For a die, the probability of rolling an even
number N(S) = 6 (6 total outcomes in the sample space) N(E) = 3 (3 outcomes for the event) P(E) = 3/6 or 1/2
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EXAMPLE
● A more complex example● Three students (Katherine, Michael, and
Dana) want to go to a concert but there are only two tickets available
● Two of the three students are selected at random What is the sample space of who goes? What is the probability that Katherine goes?
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TREE DIAGRAM
● Example continued● We can draw a tree diagram to solve
this problem● Who gets the first ticket? Any one of
the three… Katherine
Michael
Dana
Start
First ticket
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TREE DIAGRAM
● Who gets the second ticket? If Katherine got the first, then either Michael or
Dana could get the second Michael
Dana
Katherine
Michael
Dana
Start
First ticket
Second ticket
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EXAMPLE
That leads to two possible outcomes
Michael
Dana
Second ticket
Katherine
Michael
Dana
Start
First ticket
KatherineMichael
KatherineDana
Outcomes
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EXAMPLE
We can fill out the rest of the tree What’s the ProbabilityThat Katherine Gets a ticket?
KatherineMichael
KatherineDana
MichaelKatherine
MichaelDanaDana
DanaKatherine
DanaMichael
Katherine
Michael
Katherine
Michael
Dana
Start
Katherine
Michael
Dana
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SUBJECTIVE PROBABILITY
● A subjective probability is a person’s estimate of the chance of an event occurring
● This is based on personal judgment● Subjective probabilities should be
between 0 and 1, but may not obey all the laws of probability
● For example, 90% of the people consider themselves better than average drivers …
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SUMMARY
Probabilities describe the chances of events occurring … events consisting of outcomes in a sample space
Probabilities must obey certain rules such as always being greater than or equal to 0
There are various ways to compute probabilities, including empirically, using classical methods, and by simulations