Binomial Distribution: From Russia at War

The significance of the binomial distribution is its numerous applications since it is the center of a binary situation with only two possible outcomes. Many random events worth studying have only two outcomes. Most particularly, this happen when we evaluate a sample from a large population of “units” for the occurrence of a characteristics that either exhibit or not. The population are often people, event, or investment or any other number of possibilities.

The binomial distribution is used in statistics as a discrete distribution opposite to a normal distribution or continuous distribution. Binomial distribution depict only two characteristics such as “1” code for success or “0” for otherwise or failure given the number of trials in the population. Thus the binomial distribution represents the probability for success in n trials given an outcome for each event.

The numbers of events or observations are summarized in a binomial distribution when each event has the same probability of one specific outcome. The probability of observing a specific number of success outcomes in a specific number of events are determined in the binomial distribution.

The formula for binomial distribution is

Wherein...

b is the binomial probability; x is the total number of success (1); P is the success probability of each event and n is the number of trials.

The Excel spread sheet has the binomial function (BINOM.DIST) which computes the binomial distribution probability of success based on the specific number of events. As a statistical measure binomial distribution is used to determine the probability of a specified number of successes happening from an indicated number of independent trials based on the two forms used:

The Probability Mass Function (TRUE) in Excel, compute the probability of number of success (x) from the number of independent events (n)

The Cumulative Distribution Function (FALSE) in Excel, compute the probability being at most number success (x) from independent events (n).

For instance, in a store, there were 90 customers who entered and 78 customers who bought in the grocery. Based on the data the probability is 82 percent of the customer who purchase items in the grocery. What is the probability of cumulative distribution function (cdf) and the probability mass function (pmf).

The figure below show the Excel BINOM.DIST function

The binomial distribution can be used although naively to compute the probability of Russia winning the war with Ukraine. Over the last seven (7) wars involving Russia, they won 6 or 86% and lost 1 or 14%. What is the probability that Russia will win over Ukraine?

Outcome Frequency Percent

Loss (0) 1 14.29

Won (1) 6 85.71

Total 7 100

Depicted below is the cdf and the pmf

Statistically depicted, Russia will win. What do you think?