Binomial Distribution

We now develop the binomial probability distribution that is used extensively in many applied business and economic problems. Our approach begins by first developing the Bernoulli model, which is a building block for the Binomial. We consider a random experiment that can be give rise to just two possible mutually exclusive and collectively exhaustive outcomes,  which for convenience we will label “success” and “failure”. Let p denote the probability of success, so that the probability of failure is (1 – p). Now define the random variable X so that X takes the value 1 if the outcome of the experiment is success and 0 otherwise. The probability function of this random variable is then

                 P(X=0) = (1 – p)     and   P(X=1) = p

This distribution is known as Bernoulli distribution.

The Binomial Distribution

Suppose that a random experiment can result in two possible mutually exclusive and collectively exhaustive outcomes, “success” and “failure,” and that P is the probability of a success in a single trial. If n independent trials are carried out, the distribution of the number of resulting success, x, is called the binomial distribution. Its probability distribution function for the binomial random variable X = x is

Mean and Variance of Binomial Distribution

Let X be the number of successes in n independent trials, each with probability of success P. Then X follows a binomial distribution with mean

Example: An insurance broker believes that for a particular contract the probability of making a sale is 0.4. Suppose that the broker has five contracts.

a.    Find the probability that she makes at most one sale.
b.    Find the probability that she makes between two and four sales (inclusive).
c.    Graph the probability distribution function.

Comments:
•    This shape is typical for binomial probability when P is neither very large nor very small.
•    At least extremes (o or 5 sales), the probabilities are quite small.

Example: Early in August an undergraduate  college discovers that it can accommodate a few extra students, Enrolling those additional students would provide a substantial increase in revenue without increasing the operating costs of the college; that is, no new classes would have to be added. From past experience the college knows that 40% of those students admitted will actually enroll.
a.    What is the probability that at most 6 students will enroll if the college offers admission to 10 more students?
b.    What is the probability that more than 12 will actually enroll if admission offered to 20 students?
c.    If 70% of those students admitted actually enroll, what is the probability that at least 12 out of 15 students will actually enroll?

Solution:
a.    This probability can be obtained using the cumulative binomial probability distribution from Table 3 in the Appendix. The probability of at most 6 students enrolling if n = 10 and P = 0.40 is

             
     
b.    The probability that at least 12 out of 15 students enroll is the same as the probability that at most 3 out of 15 students do not enroll (the probability of a student not enrolling is 1 – 0.70 = 0.30).

            
             
     

The Poisson Probability Distribution

The Poisson probability distribution is an important discrete probability distribution for a number of applications, including:
1.    the number of failures in a large computer system during a given day.
2.    The number of replacement orders for a part received by a firm in a given month.
3.    The number of dents, scratches, or other defects in a large roll of sheet metal used to manufacture filters.

Assumptions of the Poisson Probability Distribution

Assume that an interval is divided into a very large number of subintervals so that the probability of the occurrence of an event in any subinterval is very small. The assumptions of a Poisson probability distribution are as follows:
1.    The probability of the occurrence of an event is constant for all subintervals.
2.    There can be no more than one occurrence in each subinterval.
3.    Occurrences are independent; that is, the occurrences in nonoverlapping intervals are independent of one another. 
      
We can derive the equation for computing Poisson probabilities directly from the binomial probability distribution by taking the mathematical limits as. With these limits the parameter is a constant that specifies the average number of occurrences (success) for a particular time and/or space.

The Poisson Probability Distribution Function, Mean and Variance

The random variable X is said to follow the Poisson probability distribution if it has the probability function

Example: A computer center manager, reports that his computer system experienced three component failures during the past 100 days.

a.    What is the probability of no failures in a given day?
b.    What is the probability of one or more component failures in a given day?
c.    What is the probability of at least two failures in a 3-day period?

Solution: A modern computer system has a very large number of components, each of which could fail and thus result in a computer system failure. To compute the probability of failures using the Poisson distribution, assume that all each of the millions of components has the same very small probability of failure. Also assume the first failure does not affect the probability of a second failure (in some cases, these assumptions may not hold, and more complex distributions would be used).

    
Example: Customers arrive at a photocopying machine at an average rate of two every 5 minutes. Assume that these arrivals are independent, with a constant arrival rate, and that this problem follows a Poisson model, with X denoting the number of arriving customers in a 5-minute period and mean  = 2. Find the probability that more than two customers arrive in a 5-minute period.

Solution: Since the mean number of arrivals in 5 minutes is two, then  = 2. To find the probability that more than two customers arrive, first compute the probability of at most two arrivals in a 5-minute period, and then use complement rule.

Poisson Approximation to the Binomial Distribution

Previously, we noted that the Poisson probability distribution is obtained by starting with the Binomial probability distribution with P approaching 0 and n becoming very large. Thus, it follows that the Poisson distribution can be used to approximate the binomial probabilities when the number of trials, n is large and at the same time the probability, P, is small (generally such that)

Example: An analysts predicted that 3.5% of all small corporations would file for bankruptcy in the coming year. For a random sample of 100 small corporations, estimate the probability that at least 3 will file for bankruptcy in the next year, assuming that the analyst’s prediction is correct.

Solution: The distribution of X, the number of fillings for bankruptcy, is binomial with n = 100 and P = 0.035, so that the mean of the distribution is . Using the Poisson distribution to approximate the probability of at least 3 bankruptcies, we find

The Poisson probability is simply an estimate of the actual binomial probability.