if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value
The probability of any chance process is a number between which 2 numbers
0 to 1
what does a probability 0 mean?
the outcome never occurs, no chance of occuring
what does probability 1 mean?
the outcome always occurs, 100% chance
does probability allow us to make short run predictions?
no - we can only make long run predictions with probability, as it is not very accurate in the short run. Random behavior evens out in the long run
are future outcomes affected by past behavior?
no - past outcomes do not influence the likelihood of individual outcomes occurring in the future
sample space (s)
of a chance process, this is the set of all possible outcomes
what does a probability model consist of?
2 parts: a sample space s and a probability for each outcome
any collection of outcomes from some chance process. that is, and event is a subset of the sample space
do all the possible outcomes in a probability model have to add up to one?
yes- no more, no less, if s is the sample space, p(s)=1
if 2 events have no outcomes in common, how do you find the probability that one or the other occurs?
find the sum of their individual probabilities
when are two events mutually exclusive, or disjoint?
is they have no outcomes in common, and so they can never occur together. if A and B are mutually exclusive, p(A or B)= p(A) + p(B) (imagine a split apart venn diagram)
Mutually Exclusive or not?: a single card is chosen at random from a standard deck of playing cards. what is the probability of choosing a 5 or king?
Mutually Exclusive or not?: a single card is chosen at random from a standard deck of playing cards. what is the probability of choosing a club or king?
not mutually exclusive
general addition rule for the probability of 2 events (including events that are not mutually exclusive)
if A and B are any 2 events resulting from some chance process, then p(A or B) = p(A) + p(B) - p(A and B)
what does the complement A^c contain?
exactly the outcomes that are not in A
what does AUB mean
all the outcomes in A or B
What does AnB mean
all the outcomes in both events A and B
the probability that one event happens given that another event is already known to have happened. suppose that we know that event A has happened. then the probability that event B happens given that event A has happened is denoted by : p(A|B) ->
when the chance that event B occurs is not affected by whether event A occurs
How do you prove mathematically that 2 events are independent?
events A and B are independent if: p(A|B)=p(A) and p(B|A)=p(B)
multiplication rule for independent events
if A and B are independent events, then the probability that A and B both occur is: p(AnB)=p(A)*p(B)
Conditional Probability formula
Mutually Exclusive vs independence
mutually exclusive: 2 events cannot happen together Independent Event: the occurrence of one event does not affect the occurrence of the other
What do you do after you have found the probability?
ALWAYS restate the probability in the context of the problem
how do you do at least one problems?
-find the probability of the opposite occurring. -take this probability to the power of how many people or number of tests are needed (ex: 200 tests = p(opposite occurring)^200) -then to get the probability back from the opposite, subtract this number from 1
a ____________ describes chance behavior by listing the possible outcomes in the _____________ and giving the probability that each outcome occure
probability model, sample space s
imitation of chance behavior, often carried out with random numbers
what is the complement equation
addition rule for mutually exclusive events
if A and B are disjoint, p(A or B)=p(A)+p(B)
AuB - all outcomes in A, B, or Both
AnB - consists of outcomes in both A and B
can two events be independent if they are mutually exclusive?