1. Construction Engineering 221 Probability and Statistics Binomial Distributions

2. 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

3. 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)

4. 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)! *p X q n-X

5. 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 10 12 /8.7178X10 10 *.2288 *.1 =1.5 *.2288 *.1 =.0343 or 3.4%

6. 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): ˆ P = X/n Formula as a proportion is same

7. 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

8. Binomial Distribution Binomial probabilities can be represented by tree diagrams (P=pass, F=fail) P F.9.1 P PP P P P F F F F F F.9.1.729.0810.081.009.081.009.001 Sum of all probabilities must equal 1.0