Transcript Prob & binomial dist I

Construction Engineering 221
Probability and Statistics
Binomial Distributions
Binomial Distribution
• The binomial distribution is a discrete
probability distribution
• The binomial distribution is appropriate for
modeling decision-making situations
• There can only be TWO possible
outcomes (yes/no; pass/fail)
• Outcomes have to be mutually exclusive
• Outcomes in the series of trials are
independent events
Binomial Distribution
• Probability of success (yes, pass) is
denoted by constant p (stationary process)
• Process is called Bernoulli process
• Binomial distribution used to determine
probability of obtaining a designated
number of successes (passes)
Binomial Distribution
• Three inputs required:
– Designated number of passes X
– Number of trials or observations n
– Probability of pass for each trial p
– Probability of fail is q = (1-p)
• Probability of a designated number of
passes is denoted as P(X/n,p) =
n!/X!(n-X)! *pXqn-X
Binomial Distribution
• Example- The probability that a randomly
chosen weld will pass inspection is .90. If
an inspector checks 15 welds, the
probability that exactly 14 will pass is:
P(X=14/n=15, p=.9) =
15!/14!(15-14)! * (0.9)14(0.1)1 =
1.3077 X 1012/8.7178X1010 * .2288 *.1
=1.5 *.2288 * .1 = .0343 or 3.4%
Binomial Distribution
• To determine the cumulative probability (X
or more; X or fewer), the probability of
each outcome must be determined and
the probabilities summed
• The binomial variable X can be expressed
as a proportion (percentage of passes):
ˆP = X/n
Formula as a proportion is same
Binomial Distribution
• When number of observations n is large,
and probabilities p or q are small, the
Poisson distribution is a suitable
approximation or the binomial distribution
• Rule of thumb is n>30, and n*p or n*q<5
• When n is large, but n*p or n*q is less than
5, the normal distribution is a close
approximation of binomial probabilities
Binomial Distribution
• Binomial probabilities can be represented
by tree diagrams (P=pass, F=fail)
.9
P
P
.9
.1
P
.9
F
.9
.1
.9
P
F
.9
F
.1
.1
F
F .0810
P
.1
.729
.081
.009
P
.081
F
.009
P
.009
.9
.1
.1
F
.001
Sum of all
probabilities must
equal 1.0