Chapter 5 Stats
probability of outcomes that never occur
0
the probability of any event is a number between
0 and 1 (for any event A, 0≤(A)≤1)
probability of outcomes that happen on every repetition
1
all possible outcomes together must have probabilities that add up to
1 (P(S)=1)
the probability that an event does not occur is
1 minus the probability that the event does occur (P(A^c)=1-P(A))
complement of A
A^c; event "not A"
general multiplication rule
P(A and B)=P(A ∩ B)=P(A)*P(B | A) —> in other words, for both of two events to occur, first one must occur and then given that the first event has occurred, the second must occur
P(A | B) formula
P(A | B)=P(A ∩ B)/P(B)
formula for probability of event A if all outcomes in the sample space are equally likely
P(A)=number of outcomes corresponding to event A/total number of outcomes in sample space
probability model
a description of some chance process that consists of two parts; a sample space S and a probability for each outcome
the probability of any outcome of a chance process
a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions
event
any collection of outcomes from some chance process; a subset of the sample space; usually designated capital letters (like A, B, etc.)
why don't short runs of random phenomena like tossing coins or shooting a basketball look random to us sometimes?
because they do not show the regularity that emerges only in very many repetitions; the phrase that someone is "due" because a certain outcome hasn't happened in a while is incorrect --> the next repetition does not depend on the previous outcome
Venn diagrams with non-mutually exclusive events
draw two circles that overlap with a rectangle around them; non-overlapping parts represent one event happening but not the other, the overlap represents both happening; the space outside the circles but inside the rectangle represents neither event happening
intersection
event "A and B"; A (upside U) B
union
event "A or B"; A U B
multiplication rule for independent events
if A and B are independent events, then the probability that A and B both occur is P(A ∩ B)=P(A)*P(B)
the law of large numbers
if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a specific outcome
disjoint
mutually exclusive; two events A and B have no outcomes in common and so they can never occur together —> P(A and B)=0
two events A and B are independent if
the occurrence of one event does not change the probability that the other event will happen; ie. events A and B are independent if P(A | B)=P(A) and P(B | A)=P(B) OR if it obeys the general multiplication rule for independent events
parameters for whether something is a legitimate probability model
the probability of each outcome is a number between 0 and 1 and the probabilities of all the possible outcomes add to 1
P(A)
the probability of event A
conditional probability
the probability that one event happens given that another event is already known to have happened —> if we know that event A has happened, then the probability that event B happens given that event A has happened is P(B | A)
sample space (s)
the set of all possible outcomes [of a chance process]
if two events are mutually exclusive the probability that one or the other occurs
the sum of their individual probabilities (P(A or B)=P(A)+P(B), when mutually exclusive)
tree digram
to display the sample space when a chance process involves a sequence of outcomes; the first set of branches displays the probabilities of the first events (write the event in a box and draw arrows to them from a single point with the probabilities labeled on the arrows); the second set of branches are conditional probabilities given that the first event has occurred (again events in boxes, probabilities (the conditional ones) labeled on arrows) —> the total probability of each event in the second branch occurring is found by multiplying the probability of the two arrows that lead to it
true or false: chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run
true!
true or false: two mutually exclusive events can never be independent
true! because if one event happens, the other event is guaranteed not to happen
Venn diagram for mutually exclusive events
two circles that don't overlap
general addition rule
when A and B are any two events resulting from some chance process then P(A or B)=P(A)+P(B)-P(A and B) OR WRITTEN AS P(A U B)=P(A)+P(B)-P(A ∩ B)
simulation
the imitation of chance behavior, based on a model that accurately reflects the situation
performing a simulation
state: ask a question of interest about some chance process plan: describe how to use a chance device to imitate one repetition of the process; tell what you will record at the end of each repetition do: perform many repetitions of the simulation conclude: use the results of your simulation to answer the question of interest
going in reverse using general multiplication rule
suppose that the 2nd event occurs in a sequence, what is the probability that one of the first events occurred —> P(first event | second event)=P(first event and second event)/P(second event)
