1. Construction Engineering 221 Statistics and Probability Binomial Distribution Part II

2. Binomial Distribution Binomial probability distributions use combination factorials (if order is of no importance, PPPFP) or permutations is order is of importance (PPPPF) to calculate probability of events. Combinations are just series of permutations. The distribution is used in quality control and other Bernoullian processes

3. Binomial Distribution Binomial probability tables (handout) help simplify the analysis If pass rate is 0.9, may have to find the probability of failure pattern (failure = 0.1) and subtract from one because most binomial tables only go to p = 0.5 Probability for each pattern of outcomes can be depicted as a tree diagram (Fig 5- 3, page 53)

4. Binomial Distribution Outcomes can also be shown in tabular form (Table 5-2, page 54) The mean of a binomial distribution is denoted by µ: µ = n*π, where µ is the expected value of X (in order words, the most common outcome in the distribution), n is the number of trials, and π is the probability success for each independent event

5. Binomial Distribution Example- if we flip a coin 10 times, the mean of the binomial probability distribution is 5 (10 *.5)- the “expected value” of number of heads is five Example from craps table- role of two dice has an expected value of 2(1/36) + 3(2/36)…..11(2/36) + 12(1/36) = 7- which is the number that takes all the money off the board (“crap out”)

6. Binomial Distribution The variance of a binomial distribution: Σ(x-μ) 2 *P(x), or nπ(1-π) 10 flips of a coin has a mean (expected value) of 5 heads with a variance of 10*.5*.5, or 2.5. standard deviation is square root of 2.5, or 1.6

7. Binomial Distribution Do exercise 14 in class and check answers in back.