Probability
A
manager frequently faces situations in which neither classical nor empirical
probabilities
are useful. For example, in a one-shot situation such as the launch of a
unique
product, the probability of success can neither be calculated nor estimated
from
repeated
trials. However, the manager may make an educated guess of the
probability.
This subjective probability can be thought of as a person's degree of
confidence
that the event will occur. In absence of better information upon which to rely,
subjective
probability may be used to make logically consistent decisions, but the
quality
of those decisions depends on the accuracy of the subjective estimate.
1)
Consider the following
Venn diagram in which each of the 25 dots represents an
outcome and each of the
two circles represents an event.
In the above diagram, Event A is considered to have
occurred if an experiment's
Outcome, represented by one of the dots, falls
within the bounds of the left circle.
Similarly, Event B is considered to have occurred if
an experiment's outcome falls
within the bounds of the right circle. If the
outcome falls within the overlapping region of
the two circles, then both Event A and Event B are
considered to have occurred.
There are 5 outcomes that fall in the definition of
Event A and 6 outcomes that fall in the
definition of Event B. Assuming that each outcome
represented by a dot occurs with
equal probability, the probability if Event A is
5/25 or 1/5, and the probability of Event B
is 6/25. The probability of Event A or Event B would
be the total number of outcomes in
the orange area divided by the total number of
possible outcomes. The probability of
Event A or Event B then is 9/25.
Note that this result is not simply the sum of the
probabilities of each event, which
would be equal to 11/25. Since there are two
outcomes in the overlapping area, these
outcomes are counted twice if we simply sum the
probabilities of the two events. To
prevent this double counting of the outcomes common
to both events, we need to
subtract the probability of those two outcomes so
that they are counted only once. The
result is the law of addition, which states that the
probability of Event A or Event B (or
both) occurring is given by:
P(A
or B) = P(A) + P(B) - P(A and B)
This addition rule is useful for determining the
probability that at least one event will
occur. Note that for mutually exclusive events there
is no overlap of the two events so:
P(A and B) = 0
and the law of addition reduces to:
P(A
or B) = P(A) + P(B)
2)
Sometimes it is useful
to know the probability that an event will occur given that another
event occurred. Given
two possible events, if we know that one event occurred we can
apply this information
in calculating the other event's probability. Consider the Venn
diagram of the previous
section with the two overlapping circles. If we know that Event
B occurred, then the
effective sample space is reduced to those outcomes associated
with Event B, and the
Venn diagram can be simplified as shown:
The probability that
Event A also has occurred is the probability of Events A and B
relative to the
probability of Event B. Assuming equal probability outcomes, given two
outcomes in the
overlapping area and six outcomes in B, the probability that Event A
occurred would be 2/6.
More generally,
P(A given B) = P(A and B)
----------------------------
P(B)
3) Law
of Multiplication
The probability of both
events occurring can be calculated by rearranging the terms in
the expression of
conditional probability. Solving for P(A and B), we get:
P(A
and B) = P(A given B) x P(B)
For independent events,
the probability of Event A is not affected by the occurance of
Event B, so P(A given
B) = P(A), and
P(A
and B) = P(A) x P(B)
