Chapter 4
The Binomial Distribution
In This Chapter
▶ Identifying a binomial random variable
▶ Finding probabilities using a formula or table
▶ Calculating the mean and variance
A
random variable is a characteristic, measurement, or
count that changes randomly according to some set
of probabilities; its notation is X, Y, Z, and so on. A list of all
possible values of a random variable, along with their probabilities is called a probability distribution. One of the most
well-known probability distributions is the binomial. Binomial
means “two names” and is associated with situations involving two outcomes: success or failure (hitting a red light or not;
developing a side effect or not). This chapter focuses on the
binomial distribution —when you can use it, finding probabilities for it, and finding the expected value and variance.
Characteristics of a Binomial
A random variable has a binomial distribution if all of following conditions are met:
1. There are a fixed number of trials (n).
2. Each trial has two possible outcomes: success or
failure.
3. The probability of success (call it p) is the same for
each trial.
4. The trials are independent, meaning the outcome of
one trial doesn’t influence that of any other.
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Statistics Essentials For Dummies
Let X equal the total number of successes in n trials; if all of
the above conditions are met, X has a binomial distribution
with probability of success equal to p.
Checking the binomial conditions
step by step
You flip a fair coin 10 times and count the number of heads.
Does this represent a binomial random variable? You can
check by reviewing your responses to the questions and statements in the list that follows:
1. Are there a fixed number of trials?
You’re flipping the coin 10 times, which is a fixed
number. Condition 1 is met, and n = 10.
2. Does each trial have only two possible outcomes —
success or failure?
The outcome of each flip is either heads or tails, and
you’re interested in counting the number of heads, so
flipping a head represents success and flipping a tail is
a failure. Condition 2 is met.
3. Is the probability of success the same for each trial?
Because the coin is fair the probability of success (getting a head) is p = 1⁄2 for each trial. You also know that
1 – 1⁄2 = 1⁄2 is the probability of failure (getting a tail) on
each trial. Condition 3 is met.
4. Are the trials independent?
We assume the coin is being flipped the same way
each time, which means the outcome of one flip
doesn’t affect the outcome of subsequent flips.
Condition 4 is met.
Non-binomial examples
Because the coin-flipping example meets the four conditions,
the random variable X, which counts the number of successes
(heads) that occur in 10 trials, has a binomial distribution with
n = 10 and p = 1⁄2. But not every situation that appears binomial
actually is binomial. Consider the following examples.
Chapter 4: The Binomial Distribution
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No fixed number of trials
Suppose now you are to flip a fair coin until you get four
heads, and you count how many flips it takes to get there.
(That is, X is the number of flips needed.) This certainly
sounds like a binomial situation: Condition 2 is met since
you have success (heads) and failure (tails) on each flip;
Condition 3 is met with the probability of success (heads)
being the same (0.5) on each flip; and the flips are independent, so Condition 4 is met.
However, notice that X isn’t counting the number of heads,
it counts the number of trials needed to get 4 heads. The
number of successes (X) is fixed rather than the number of
trials (n). Condition 1 is not met, so X does not have a binomial distribution in this case.
More than success or failure
Some situations involve more than two possible outcomes yet
they can appear to be binomial. For example, suppose you roll
a fair die 10 times and record the outcome each time. You have
a series of n = 10 trials, they are independent, and the probability of each outcome is the same for each roll. However, you’re
recording the outcome on a six-sided die. This is not a success/
failure situation, so Condition 2 is not met.
However, depending on what you’re recording, situations
originally having more than two outcomes can fall under
the binomial category. For example, if you roll a fair die 10
times and each time record whether or not you get a 1, then
Condition 2 is met because your two outcomes of interest are
getting a 1 (“success”) and not getting a 1 (“failure”). In this
case p = 1/6 is the probability for a success and 5/6 for failure.
This is a binomial.
Probability of success (p) changes
You have 10 people — 6 women and 4 men — and form a committee of 2 at random. You choose a woman first with probability 6/10. The chance of selecting another woman is now 5/9.
The value of p has changed, and Condition 3 is not met. This
happens with small populations where replacing an individual
after they are chosen (to keep probabilities the same) doesn’t
make sense. You can’t choose someone twice for a committee.
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Statistics Essentials For Dummies
Trials are not independent
The independence condition is violated when the outcome
of one trial affects another trial. Suppose you want to know
support levels of adults in your city for a proposed casino.
Instead of taking a random sample of say 100 people, to save
time you select 50 married couples and ask each individual
what their opinion is. Married couples have a higher chance
of agreeing on their opinions than individuals selected at
random, so the independence Condition 4 is not met.
Finding Binomial Probabilities
Using the Formula
After you identify that X has a binomial distribution (the four
conditions are met), you’ll likely want to find probabilities
for X. The good news is that you don’t have to find them from
scratch; you get to use previously established formulas for
finding binomial probabilities, using the values of n and p
unique to each problem.
Probabilities for a binomial random variable X can be found
using the formula
, where
✓ n is the fixed number of trials.
✓ x is the specified number of successes.
✓ n – x is the number of failures.
✓ p is the probability of success on any given trial.
✓ 1 – p is the probability of failure on any given trial. (Note:
Some textbooks use the letter q to denote the probability
of failure rather than 1 – p.)
These probabilities hold for any value of X between 0 (lowest
number of possible successes in n trials) and n (highest
number of possible successes).
The number of ways to arrange x successes among n trials is
called “n choose x,” and the notation is
. For example,
means “3 choose 2” and stands for the number of ways to get
2 successes in 3 trials. In general, to calculate “n choose x,”
Chapter 4: The Binomial Distribution
you use the formula
39
. The notation n! stands
for n-factorial, the number of ways to rearrange n items. To
calculate n!, you multiply n(n – 1)(n – 2) . . . (2)( 1). For example 3! is 3(2)(1) = 6; 2! is 2(1) = 2; and 1! is 1. By convention, 0!
equals 1. To calculate “3 choose 2,” you do the following:
Suppose you cross three traffic lights on your way to work,
and the probability of each of them being red is 0.30. (Assume
the lights are independent.) You let X be the number of red
lights you encounter and you want to find the probability
distribution for X. You know p = probability of red light =
0.30; 1 – p = probability of a non-red light = 1 – 0.30 = 0.70; and
the number of non-red lights is 3 – X. Using the formula, you
obtain the probabilities for X = 0, 1, 2, and 3 red lights:
The final probability distribution for X is shown in Table 4-1.
Notice they all sum to 1 because every possible value of X is
listed and accounted for.