1. Applied Statistical and Optimization Models
Topic 06-03: The Binomial Distribution

2. Objectives Assumptions Discrete variable Bernoulli Experiment
Binomial distributions generated in Excel

3. Probability The Binomial Distribution
The binomial distribution is a discrete probability distribution that is applicable as a model for decision- making situations in which a sampling process can be assumed to conform to a Bernoulli process. A Bernoulli process is subject to the following assumptions:
Assumptions:
1. Only two mutually exclusive possible outcomes are possible in each trial, or observation. For convenience these are called success and failure.
2. The outcomes in the series of trials, or observations, constitute independent events.
3. The probability of success in each trial, denoted by p, remains constant from trial to trial. That is, the process is stationary.
Daniel Bernoulli

4. Probability
Example: Find the probability of obtaining two heads in six tosses of a fair coin.
Then: X = 2, n=6, p=0.5, and q=0.5

5. Probability
Because use of the binomial formula involves considerable arithmetic when the sample is relatively large, tables of binomial probabilities are often used.
Cumulative Binominal Table (N=6) n x
0.1
0.2
0.3
0.4
0.5 6
0.5314
0.2621
0.1177
0.0467
0.0156 1
0.8857
0.6554
0.4202
0.2333
0.1094 2
0.9842
0.9011
0.7443
0.5443
0.3438 3
0.9987
0.9830
0.9295
0.8208
0.6563 4
0.9999
0.9984
0.9891
0.9590
0.8906 5
1.0000
0.9993
0.9959
0.9844

6. Probability
Exercise: Create a frequency distribution for the following. What’s the probability of getting zero heads when you toss a fair coin 10 times? What’s the probability of getting 1 head? Two heads? … Ten heads? Use the Excel Formula “= Binom.Dist()”
Discuss: How would the success probability affect the skewness of the distribution?
Note: The mean of a binomial distribution is mean = n×p The variance of a binomial distribution var = n×p×(1-p) The standard deviation of a binomial distribution is sd = [n×p×(1-p)]0.5