counting and probability
joint probability
Joint probability is the probability that both events happen at the same time. -Joint probability is computed by multiplying the individual event probabilities together
if a firm is going to create three teams of four from twelve employees. Which approach is the most appropriate for determining how the twelve employees can be selected for the three teams?
Labeling formula. This problem is a labeling problem where the 12 employees will be assigned one of three labels. It requires the labeling formula. In this case there are [(12!) / (4!4!4!)] = 34,650 ways to group the employees.
We calculate the probability that at least one of the events happening using the addition rule for probabilities:
P(A or B) = P(A) + P(B) - P(A and B), where P(A and B) = P(A) × P(B)
The multiplication rule of probability is stated as:
P(AB) = P(A|B) × P(B), where P(AB) is the joint probability of events A and B.
if two events are mutually exclusive, the probability that they both will occur at the same time is
Zero. -If two events are mutually exclusive, it is not possible to occur at the same time.
A positive covariance indicates
a positive linear relationship but nothing else. The magnitude of the covariance by itself is not informative with respect to the strength of the relationship
Each lottery ticket discloses the odds of winning. These odds are based on
a priori probability
An event is
a set of one or more possible values of a random variable
An empirical probability is one that is
derived from analyzing past data
For a given corporation, which of the following is an example of a conditional probability? The probability the corporation's:
dividend increases given its earnings increase.
After repeated experiments, the average of the outcomes should converge to?
expected value. It is the long-run average of all outcomes.
A priori probability is based on
formal reasoning and inspection, rather than on historical results or subjective opinion. -ex. On a random draw, the probability of choosing a stock of a particular industry from the S&P 500.
The joint probability of independent events
is obtained by multiplying the probabilities of the individual events together
If event A and event B cannot occur simultaneously, then events A and B are said to be
mutually exclusive
If the outcome of event A is not affected by event B, then events A and B are said to be
statistically independent
A set of probabilities must
sum to one.
For the task of arranging a given number of items without any sub-groups, this would require
the factorial function The factorial function, denoted n!, tells how many different ways n items can be arranged where all the items are included.
The labeling formula determines
the number of different ways to assign a given number of different labels to a set of objects. nLx^g= n!/(x!)^g
the combination formula is used to find
the number of possible ways to draw r objects from a set of n objects when the order doesn't matter. nCx= n!/x!(n-x)!
The permutation formula is used to find
the number of possible ways to draw r objects from a set of n objects when the order in which the objects are drawn matters. nPx= n!/(n-x)!
An outcome is
the numerical result associated with a random variable.
Conditional probability is
the probability of one event happening given that another event has happened.
The unconditional probability of an event, given conditional probabilities, is determined by using the:
the total probability rule= P(A) = P(A|B1)P(B1) + P(A|B2)P(B2) + ... + P(A|Bn)P(Bn)
The multiplication rule of counting is used
to determine the number of different ways to choose one object from each of two or more groups.
If two events are independent, the probability that they both will occur is:
Cannot be determined from the information given.
A company says that whether it increases its dividends depends on whether its earnings increase. From this we know
P(earnings increase | dividend increase) is not equal to P(earnings increase).
what are the odds for the event occurring?
The answer can be determined by dividing the probability of the event by the probability that it will not occur
With respect to the units each is measured in, which of the following is the most easily directly applicable measure of dispersion?
The standard deviation is in the units of the random variable itself and not squared units like the variance.
The total probability rule is used to
calculate the unconditional probability of an event from the conditional probabilities of the event given a mutually exclusive and exhaustive set of outcomes
the covariance can be
can be positive or negative.
