Binomial Formula: Overview
A binomial is just an experiment that can have two outcomes: success or failure. For example, a coin toss can be a binomial experiment. If you wanted to know how many heads would come up if you tossed the coin three times, the “success” would be getting a heads on a single toss.
Although the concept of a binomial is simple, the binomial formula–used to calculate the probability of success for binomial distributions–isn’t the easiest of formulas to work . If you use a TI-83 or TI-89, much of the work is done for you. However, if you don’t own one, here are the steps you should follow to get the answer right every time.
Note: The ! symbol is a factorial (What is a factorial?)
Sample question: “60% of people who purchase sports cars are men. If 10 sports car owners are randomly selected, find the probability that exactly 7 are men.”
Step 1:: Identify ‘n’ and ‘X’ from the problem. Using our sample question, n (the number of randomly selected items — in this case, sports car owners are randomly selected) is 10, and X (the number you are asked to “find the probability” for) is 7.
Step 2: Figure out the first part of the formula, which is:
n! / (n – X)! X!
Substituting the variables:
10! / ((10 – 7)! × 7!)
Which equals 120. Set this number aside for a moment.
Step 3: Find “p” the probability of success and “q” the probability of failure. We are given p = 60%, or .6. therefore, the probability of failure is 1 – .6 = .4 (40%).
Step 4: Work the next part of the formula.
Set this number aside while you work the third part of the formula.
Step 5: Work the third part of the formula.
q(.4 – 7)
Step 6: Multiply the three answers from steps 2, 4 and 5 together.
120 × 0.0279936 × 0.064 = 0.215.
If you’re interested in a more technical rundown on the binomial formula, check out this article.