The binomial distribution formula is:
b(x; n, P) = nCx * Px * (1 – P)n – x
Where:
b = binomial probability
x = total number of “successes” (pass or fail, heads or tails etc.)
P = probability of a success on an individual trial
n = number of trials
Note: The binomial distribution formula can also be written in a slightly different way, because nCx = n! / x!(n – x)! (this binomial distribution formula uses factorials (What is a factorial?). “q” in this formula is just the probability of failure (subtract your probability of success from 1).
Using the First Binomial Distribution Formula
The binomial distribution formula can calculate the probability of success for binomial distributions. Often you’ll be told to “plug in” the numbers to the formula and calculate. This is easy to say, but not so easy to do—unless you are very careful with order of operations, you won’t get the right answer. If you have a Ti-83 or Ti-89, the calculator can do much of the work for you. If not, here’s how to break down the problem into simple steps so you get the answer right—every time.
Example 1
Q. A coin is tossed 10 times. What is the probability of getting exactly 6 heads?
I’m going to use this formula: b(x; n, P) – nCx * Px * (1 – P)n – x
The number of trials (n) is 10
The odds of success (“tossing a heads”) is 0.5 (So 1-p = 0.5)
x = 6
P(x=6) = 10C6 * 0.5^6 * 0.5^4 = 210 * 0.015625 * 0.0625 = 0.205078125
How to Work a Binomial Distribution Formula: Example 2
80% of people who purchase pet insurance are women. If 9 pet insurance owners are randomly selected, find the probability that exactly 6 are women.
Step 1: Identify ‘n’ from the problem. Using our example question, n (the number of randomly selected items) is 9.
Step 2: Identify ‘X’ from the problem. X (the number you are asked to find the probability for) is 6.
Step 3: Work the first part of the formula. The first part of the formula is
n! / (n – X)! X!
Substitute your variables:
9! / ((9 – 6)! × 6!)
Which equals 84. Set this number aside for a moment.
Step 4: Find p and q. p is the probability of success and q is the probability of failure. We are given p = 80%, or .8. So the probability of failure is 1 – .8 = .2 (20%).
Step 5: Work the second part of the formula.
pX
= .86
= .262144
Set this number aside for a moment.
Step 6: Work the third part of the formula.
q(n – X)
= .2(9-6)
= .23
= .008
Step 7: Multiply your answer from step 3, 5, and 6 together.
84 × .262144 × .008 = 0.176.
Example 3
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.
pX
= .67
= .0.0279936
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)
= .4(10-7)
= .43
= .0.064
Step 6: Multiply the three answers from steps 2, 4 and 5 together.
120 × 0.0279936 × 0.064 = 0.215.
A manufacturer of metal pistons finds that on the average, 12% of his pistons are rejected because they are either oversize or undersize. What is the probability that a batch of 10 pistons will contain
(a) no more than 2 rejects?
(b) at least 2 rejects?
Let X be the binomial random variable denoting the number of metal pistons.
Let p be the probability of rejections.
Given that p = 12% = 12/100 = 0.12, q = 0.88, n = 10.
So X ~ B(0.12, 10). Hence the p.m.f of X is given by
P(X = x) = 10Cx(0.12)x(0.88)10-x (a) P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
Thus out of a batch of 10 pistons, the probability of no more than 2 rejects is 0.89131
b) P(2 ≤ X) = 1 - P(X < 2)
Thus out of 10 pistons, the probability that at least 2 will be rejected is 0.34173