The Binomial Distribution is used to tell us the probability of an event occurring a specific number of times (x) in a situation were the following are all true:

• We know the exact number of trials for the event (N)
• We know the probability of the event's occurrence (p)
• The probability of the event's occurrence does not change
• The outcome of each trial results in one of two outcomes

• The exact number of trials(N) is 4.  We will flip the coin 4 times
• The probability of the event's occurrence is 0.5
• The number of successes for which we want to know the probability is 0.  Four tails = zero heads.

The binomial distribution equation looks like this:

pr(x) =[N!/x!((N-x)!] *(p^x)(1-p)^N-x

NOBODY PANIC!  We are not really going to try to work with that formula!  MS EXCEL has a binomial distribution calculator built into it.  We just need to know how to use it.

In an open spreadsheet select a cell and then click on the function button (fx).  Select STATISTICAL from the categories drop down menu, and select BINOMDIST from the functions list box.  Click OK to open the BINOMDIST Function Arguments box.

The function arguments box for the binomial distribution.

Now we need simply to enter the arguments and let EXCEL crank out the results. 

• Number_s is the number of successes we are interested in.  I have entered 0 to calculate the probability of getting heads 0 times in 4 flips.
• Trials is the number of times we are going to flip the coin.  In this experiment 4.
• Probability_s is the probability that it will be heads.  0.5
• Cumulative is asking whether we want to aggregate the results from our selected number of successes (Number_s) and below.  We do not want to know an aggregate, just the probability of exactly zero heads in 4 flips.  So I have entered false.

That's it, after the data is entered, EXCEL gives up an answer:  0.0625.  This corresponds to the fraction 1/16.  Or, there is a 1 in 16 chance that if we flip a coin 4 times, it will be tails all four times.

What if we want to know the probability of flipping heads 3 or more times in 4 flips? That gets a bit trickier, but not too tricky for us!  This is where we use the cumulative argument set to true.  Unfortunately, it aggregates the probabilities for our selected number of successes and below.  We want to know a selected number of successes and above.  To do this we need to find the probability for the number of successes below and then subtract that number from 1. 

Open the function arguments box for the BINOMDIST function as shown above.  Enter 2 as the number of successes, 4 as the number of trials, 0.5 as the probability and set cumulative to true.  The probability that the coin will be heads 0, 1 or 2 times is 0.6875.  Subtract this from 1 to get the probability that it will be heads 3 or 4 times: 1 - 0.6875 = 0.3125.  

Read on for an article on applying the binomial distribution in sports betting.