Handbook of Civil Engineering Calculations

Calculation Procedure:

1. Compute the number of possible committees in the absence
   of any restriction
   Since the members will be of equal rank, each committee represents a combination. Use
   the relation Cnr = n\/[rl(n-r)], or C 156 = 15!/(6!9!) = (15 x 14 x 13 x 12 x U x lQ)/(6 x
   5 x 4 x 3 x 2) = 5005.


2. Compute the number of possible committees that violate
   the imposed restriction
   Assign McCarthy to the committee, but exclude Polanski. Five members remain to be se-
   lected, and 13 individuals are available. The number of such committees = C 135 =
   13!/(5!8!) = (13x 12 x U x 10 x 9)/(5 x 4 x 3 x 2) = 1287.


3. Compute the number of possible committees that satisfy
   the requirement
   By subtraction, the number of ways in which the committee can be formed = 5005 - 1287
   -3718.

Probability

If the outcome of a process cannot be predicted because it is influenced by chance, the
process is called a trial, or experiment. The outcome of a trial or set of trials is an event.
Two events are mutually exclusive if the occurrence of one excludes the occurrence of the
other. Two events are independent of each other if the occurrence of one has no effect on
the likelihood that the other will occur.
Assume that a box contains 17 objects, 12 of which are spheres. If an object is to be
drawn at random and all objects have equal likelihood of being drawn, then the probabili-
ty that a sphere will be drawn is 12/17. Thus, the probability of a given event can range
from O to 1. The lower limit corresponds to an impossible event, and the upper limit cor-
responds to an event that is certain to occur. If two events are mutually exclusive, the
probability that either will occur is the sum of their respective probabilities. If two events
are independent of each other, the probability that both will occur is the product of their
respective probabilities.
Assume that a random variable is discrete and the number of values it can assume is fi-
nite. A listing of these values and their respective probabilities is called the probability
distribution of the variable. Where the number of possible values is infinite, the probabil-
ity distribution is expressed by stating the functional relationship between a value of the
variable and the corresponding probability. Where the random variable is continuous, the
method of expressing its probability distribution is illustrated in the calculation procedure
below pertaining to the normal distribution.

Notational System

Here E = given event; X = random variable; P(E) = probability that event E will occur;
P(X 1 ) = probability that X will assume the value X 1 ; IJL and cr = arithmetic mean and stan-
dard deviation, respectively, of a probability distribution.