The Binomial Distribution
A Probability Distribution
Binomial Distribution
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Fixed number of trials—n
On each trial there is only two possible outcomes—success and failure.
Each trial is identical and independent.
On each trial, the probability of a success, p, is always the same. The
probability of a failure is 1 – probability of a success: q = 1 – p
X
Β(n,p)
n
n
P( X  x)    p x q n  x    p x (1  p) n  x , x  0,1,2...,n
 p
 p
Example 1
Suppose that we decide to record the gender of each of
the next 25 newborns born at a local hospital. What is
the chance that 15 of them are female? What is the
chance that between 10 and 15 of them are female?
Example 2
Twenty-five percent of the customers entering a grocery store
between 5 pm and 7 pm use an express checkout. Consider
five randomly selected customers. Let x be the number who
use the express checkout.
(a) What is P(x = 2)?
(b) What is P(x ≤ 1)?
(c) What is P(x ≥ 2)?
(d) What P(x ≠ 2)?
Example 3
A biology test consists of seven multiple choice questions. Each
question has five possible answers, only one of which is correct. At
least four correct answers are required to pass the test. Juan does
not know the answer to any of the questions, so, for each question,
he selects the answer at random.
a) Find the probability that Juan answers exactly four questions
correctly.
b) Find the probability that Juan passes the biology test
Binomial Distribution
If the random variable X is such that X
1.
The expected value of X is
Β(n,p) , we have
  Ex   np
2.
The mode of X is that value of x that has the largest probability.
3.
The variance of X is
 2  Varx  npq  np(1  p)
Example 4
You have a 25 question multiple choice test in history
that you did not study for (i.e., you won’t know any of the
answers). If each question has five possible answers,
how many questions can you expect to get correct by
guessing?
Example 5
In an experiment, a trial is repeated n times. The trials are
independent and the probability p of success in each trial is
constant. Let X be the number of successes in the n trials.
The mean of X is 0.4 and the standard deviation is 0.6.
a) Find p.
b) Find n.
Example 6
The probability that a car will come to a complete stop at a
particular stop sign is .4. Assuming that the next four cars to
arrive at the stop sign are independent of one another, what
is the probability that:
(a) none of these cars will come to a complete stop.
(b) all of these cars will come to a complete stop.
(c) at least one of these cars comes to a complete stop.