34 terms

probability experiment

an experiment in which we do not know what any individual outcome will be, but we do know how a long series of repetition will come out

random phenomenon

when we know what outcomes could happen, but not which particular values will happen

probability

the proportion of times that an event occurs in the long run, as a probability experiment is repeated over and over again; an event's long-run relative frequency

trial

a single attempt or realization of a random phenomenon

outcome

the outcome of a trial is the value measured, observed, or reported for an individual instance of that trial

law of large numbers

as a probability experiment is repeated again and again, the proportion of times that a given event occurs will approach its probability

sample space

contains all the possible outcomes of a probability experiment

event

an outcome or a collection of outcomes from a sample space

probability model

for a probability experiment, consists of a sample space, along with a probability for each event; if A denotes an event and P denotes probabilities, the probability of the event A is denoted P(A)

equally likely outcomes

if a sample space has n equally likely outcomes, and an event A has k outcomes, then P(A)=number of outcomes in A/number of outcomes in the sample space=k/n

fair and unfair

a fair coin or die is one for which all outcomes are equally likely; an unfair coin or die is one for which some outcomes are more likely than others

rules for the value of a probability

a probability can never be negative, and a probability can never be greater than 1; if A cannot occur, then P(A)=0; if A is certain to occur, then P(A)=1

unusual event

an event whose probability is small; the cutoff value can by any small value that seems appropriate for a specific situation, but the most commonly used value is 0.05; in other words, a probability less than this value would be considered an unusual event

Empirical Method

consists of repeating an experiment a large number of times, and using the proportion of times an outcome occurs to approximate the probability of the outcome; it does NOT give us the exact probability, but the larger the number of replications of the experiment, the more reliable the approximation will be

compound event

an event that is formed by combining two or more events

P(A or B)

=P(A occurs or B occurs or both occur)

General Addition Rule

for any two events A and B (nondisjoint),

P(A or B)=P(A)+P(B)-P(A and B)

P(A or B)=P(A)+P(B)-P(A and B)

mutually exclusive

AKA disjoint; two events are said to be mutually exclusive if it is impossible for both events to occur

the addition rule for mutually exclusive events

if A and B are mutually exclusive events (disjoint), then

P(A or B)=P(A)+P(B)

in general, three or more events are mutually exclusive if only one of them can happen; also A and B are mutually exclusive if P(A and B)=0

P(A or B)=P(A)+P(B)

in general, three or more events are mutually exclusive if only one of them can happen; also A and B are mutually exclusive if P(A and B)=0

complements

if A is any event, the complement of A is the event that A does not occur; the complement of A is denoted A°; if an event does not occur, then its complement occurs, and if an event occurs, its complement does not occur

the rule of complements

P(A°)=1-P(A)

conditional probability

a probability computed with the knowledge of additional information

unconditional probability

a probability computed without the knowledge of additional information

conditional probability equation

the conditional probability of an event B, given an event A, is denoted P(B|A); P(B|A) is the probability that B occurs, under the assumption that A occurs; we read P(B|A) as "the probability of B, given A"

the general method for computing conditional probabilities

the probability of B given A is

P(B|A)=P(A and B)/P(A)

note that we cannot compute P(B|A) if P(A)=0

when the outcomes in the sample space are equally likely, then

P(B|A)=Number of outcomes corresponding to (A and B)/Number of outcomes corresponding to A

P(B|A)=P(A and B)/P(A)

note that we cannot compute P(B|A) if P(A)=0

when the outcomes in the sample space are equally likely, then

P(B|A)=Number of outcomes corresponding to (A and B)/Number of outcomes corresponding to A

the general multiplication rule

P(A and B)=P(A)P(B|A)

or, equivalently,

P(A and B)=P(B)P(A|B);

P(B|A) means the probability of B given the probability of A

or, equivalently,

P(A and B)=P(B)P(A|B);

P(B|A) means the probability of B given the probability of A

independence

do NOT confuse independent events with mutually exclusive events; two events are independent if the occurrence of one does not affect the probability of the occurrence of the other; two events are mutually exclusive if the occurrence of one makes it impossible for the other to occur

independent

two events are independent if the occurrence of one does not affect the probability that the other event occurs; if two events are not independent, they are dependent

the multiplication rule for independent events

if A and B are independent events, then

P(A and B)=P(A)P(B)

this rule can be extended to the case where there are far more than two independent events

P(A and B)=P(A)P(B)

this rule can be extended to the case where there are far more than two independent events

sampling with replacement

when we sample two items from a population, we can replace the first item drawn before sampling the second; when doing this, it is possible to draw the same item more than once; the draws are independent

sampling without replacement

when we sample two items from a population, we leave the first item out when sampling the second one; when doing this, it is impossible to sample an item more than once; the draws are dependent

replacement

- when sampling with replacement, the sampled items are independent

- when sampling without replacement, if the sample size is less than 5% of the population, the sampled items may be treated as independent

- when sampling without replacement, if the sample size if more than 5% of the population, the sampled items cannot be treated as independent

- when sampling without replacement, if the sample size is less than 5% of the population, the sampled items may be treated as independent

- when sampling without replacement, if the sample size if more than 5% of the population, the sampled items cannot be treated as independent

solving "at least once" problems

to compute the probability that an event occurs at least once, find the probability that it does not occur at all, and subtract from 1

solving "given" problems

P(given AND other)/P(given)