statistics
a collection of procedures and principles for gathering data and analyzing information to help people make decisions when faced with uncertainty
random circumstances
-is one in which the outcome is unpredictable
-the outcome is not determined until we observe it
ex. drawing a name out of a hat
-in other cases the outcome is already determined but out knowledge of it is uncertain
ex. having a medication or a placebo ->
probability
-is a value between 0-1
-written as a fraction or a decimal
-probability -> simply is a number between 0 and 1 that is assigned to a possible outcome of a random circumstance
-total of an assigned probabilities must equal 1
relative frequency interpretation of probability
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probability & relative frequency
-for situations that we can imagine repeating many times, we define the probability of a specific outcome as the proportion of time is would occur over the long run. this also is called the relative frequency of that particular outcome
-emphasis on long r
relative frequency probability
-2 methods to determine relative frequency probability
1. involves making an assumption about the physical world
ex odds of getting heads when flipping a coin is 1/2
2. making a direct observation of how often something happens
-observing the relative fre
proportions and percentages
-relative frequency probabilities are often derived from the proportion of individuals who have a certain characteristic, or from the proportion of the time a certain outcome occurs in a random circumstance
-0 -> outcome never occurs
-1-> outcome always o
probability of observed categorical data
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personal probability interpretation
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personal probability & subjective probability
we define personal probability of an even to be the degree to which a given individual believes that the event will happen. sometimes, the term subjective probability is used because the degree of belief may be different for each individual
-personal prob
coherent probabilities
-personal probabilities must fit together in certain ways if they are to be coherent probabilities
-by coherent it means that your personal probability of one event doesn't contradict your personal probability of another
personal probabilities
-differ by individual preference
-personal probabilities assigned to unique events are not equivalent to proportions or percentages
ex it does not make sense to say that an asteroid will hit 1 out of 1000 earths in this century
simple event
a specific possible outcome
-is a unique possible outcome of a random circumstance
-(an outcome)
-out of days of the week -> day 1 or day 2 ect
sample space
the collection of all possible outcomes is called sample space for the random circumstances
-for a random circumstance is the collection of all simple events
-(all possible outcomes)
-out of days of the week -> days 1-7
event
the term event is used to describe any collection of one or more possible outcomes
-is any collection of one or more simple events in the sample space; events can be simple events or compound events. events are often written using capital letters A, B, C,
compound event
is sometimes used if an event includes at least two simple events
-is an event that includes two or more simple events
P(A)
the probability that A occurs
valid probability
the assigned probabilities must meet the following conditions
1. each probability is between 0 and 1
2. the sum of the probabilities over all possible simple events is 1. in other words, the total probability for all possible outcomes of random circumstan
equally likely simple events
- if there are X number of equally likely simple events in the sample space, then the probability is 1/X for each one to occur
complementary events
-one event is the complement of another event is the two events do not contain any of the same events and together they cover the entire sample space. for an event A, the notation A^C represents the complement of A
ex the complement of odds is evens
-the
mutually exclusive events
-two events are mutually exclusive if they do not contain any of the same simple events (outcomes). equivalent terminology is that the two events are disjoint
-2 events that do not contain any of the same simple events
-but together the do not cover the e
independent events
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dependent events
-two events are dependent if knowing that one will occur (or has occurred) changes the probability that the other occurs
-when (1/6) Alex name is drawn out of the hat the likelihood for Mary name to be drawn out of the hate went from 1/6 to 1/5
independent and dependent events
the definitions can apply either to events within the same random circumstances or to events form two separate random circumstances.
conditional probabilities
-the conditional probability of event B, given that the event A has occurred or will occur, is long-run relative frequency with which event B occurs when circumstances are such that A has occurred or will occur. this probability is written as P(B|A)
-P(B)
how to know if events A and B are independent
1. knowing whether A will occur does not change the probability that B will occur
2.if the conditional probability P(A|B) is the same ad the unconditional probability P(A), then A and B are independent events.
simple events
is a unique possible outcome of random circumstances
sample space
for a random circumstance is the collection of all simple events
compound event
is an event that includes two or more simple events
event
-an event is any collection of one or more simple events in the sample space
-events can be simple of compound events
-events are often written using capital letters A, B, C, and so on , and their probabilities are written as P(A)
complement
-one event is the complement of another event if the two events do not contain an of the same simple events and together the cover the entire sample space
-for an event A the notation is A^C represents the complement f A
mutually exclusive
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independent
two events are independent of each other if knowing that one will occur (or has occurred) does not change the probability that the other occurs
dependent
two events are dependent if knowing that one will occur (or has occurred) changes the probability that the other occurs
conditional probability
-the conditional probability of the event B, given that the event A has occurred r will occur, is long-run relative frequency with which event B occurs given that A has occurred or will occur.
-it is written as P(B|A)
probability that an event does not occur
-rule 1 in complement rule
-"not the event
-if P(A) is the probability that something will occur then 1-P(A) is the probability that it will not occur
-to find the probability of A^C, the complement of A use P(A^C) = 1-P(A)
-P(A^C) is the probability that the event A will not occur
probability that either one o both of two events happens
-rule 2: addition rule
-addition rule for "either/or/both
-to find the probability that either A or B or both happen
-rule 2a (general): P(A or B) = P(A) + P(B) - P(A and B)
-rule 2b (for mutually exclusive events): is A and B are mutually exclusive events, P(A or B) = P(A) + P(B)
-rule 2b is a special case of r
probability that two or more events occur together
-rule 3: multiplication rule
-multiplication rule for "and
-to find the probability that two events A and B both occur simultaneously or in a sequence
-rule 3a (general): P(A and B) = P(A) X P(B|A) = P(B) X P
(A|B)
-rule 3b (for independent events): is A and B are independent events, P(A and B) = P(A) X P(B)
-ext
difference between mutually exclusive and independent
-when two events are mutually exclusive and one happens, the other event cannot happen simultaneously, so it has probability 0
-when two events are independent and one happens, the probability of the other event is unaffected
(chart in book)
none and at least one
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with replacement and without replacement
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conditional probabilities and bayes' rule
-determining a conditional probability
-rule 4 (conditional probability)
-this rule is simple and algebraic restatement of Rule 3a, but it is sometimes useful to apply the following form of that rule, which gives the probability that B occurs given that A has occurred or will occur
- P(B|A) = P(A and B) / P(A)
the assignment f
Bayes' rule
-we know conditional probabilities in one direction but we want to know conditional probabilities in the other direction
-ex. the test results for a disease come back positive but what are the odds that those results are correct
-(the first probability wa
two-way table
-when conditional or joint probabilities are known for 2 events it is useful to make a hypothetical two-way table of outcomes
-for a hundred thousand people the usually result in a whole number of people in each cell
-called the hypothetical hundred thous
tree diagram
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using simulation to estimate probabilities
simulate- some probabilities are so difficult or time consuming to calculate that it is easier to simulate the situation repeatedly by using a computer or calculator and observe the relative frequency of the event of interest
-random circumstance -> n
- o
confusion of the inverse
-ex. doctors can confuse the conditional probability of cancer given a positive x-ray with the inverse, the conditional probability of a positive x-ray given that the patient has cancer
-of 100,000 women with lumps
-10,700 x-rays say positive for malignan
to determine the probability of a positive test result being accurate
1. the base rate or probability that you are likely to have the disease, without any knowledge of the rest of your test results
2. the sensitivity of the test, which is the proportion of people who correctly test positive when thy actually have the diseas
coincidence
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the gamblers fallacy
-people tend to think that the long-run frequency of an event should apply in the short run
-people tend to believe that a string of good luck will be followed by a string of bad luck which is false
-gamblers fallacy is primarily applied to independent ev
law od small numbers
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