Week 1 - Lesson 4.5 The Binomial Probability Distribution Flashcards
Example: Suppose we select 100 students from a large university campus and ask them whether they are in favor of a
certain issue that is being debated on their campus. The students are to answer with either a ’yes’ or a ’no’. Here, we
are interested in X, the number of students who favor the issue (a ’yes’). If each student is randomly selected from
the total population of the university, and the proportion of students who favor the issue is p, then the probability
that any randomly selected student favors the issue is p. The probability of a selected student who does not favor the
issue is 1 p. Sampling 100 students in this way is equivalent to tossing a coin 100 times. This experiment is an
example of a
Binomial experiment
What are the Characteristics of a Binomial Experiment
• The experiment consists of n independent, identical trials.
• There are only two possible outcomes on each trial: S (for success) or F (for failure).
• The probability of S remains constant from trial to trial. We will denote it by p. We will denote the probability
of F by q. Thus, q = 1 p.
• The binomial random variable X is the number of successes in n trials.
Suppose a university decides to give two scholarships to two students. The pool of applicants is ten students: six
males and four females. All ten of the applicants are equally qualified, and the university decides to randomly select
two. Let X be the number of female students who receive the scholarship.If the first student selected is a female, then the probability that the second student is a female is 3
9 . Here we have a
conditional probability: the success of choosing a female student on the second trial depends on the outcome of the
first trial. Therefore, the trials are not independent, and X is
Not binomial random variable
A company decides to conduct a survey of customers to see if its new product, a new brand of shampoo, will sell
well. The company chooses 100 randomly selected customers and asks them to state their preference among the new
shampoo and two other leading shampoos on the market. Let X be the number of the 100 customers who choose the
new brand over the other two.
In this experiment, each customer either states a preference for the new shampoo or does not. The customers’
preferences are independent of each other, and therefore, X is
Binomial random variable
Let’s examine an actual binomial situation. Suppose we present four people with two cups of coffee (one percolated
and one instant) to discover the answer to this question: “If we ask four people which is percolated coffee and
none of them can tell the percolated coffee from the instant coffee, what is the probability that two of the four will
guess correctly?” We will present each of four people with percolated and instant coffee and ask them to identify
the percolated coffee. The outcomes will be recorded by using C for correctly identifying the percolated coffee and
I for incorrectly identifying it. A list of the 16 possible outcomes, all of which are equally likely if none of the four
can tell the difference and are merely guessing, is shown below:
Table 4.16 on module
Using the Multiplication Rule for Independent Events, you know that the probability of getting a certain outcome
when two people guess correctly, such as CICI, is ( 1
2 )
1
2
1
2
1
2
= 1
16
. The table shows six outcomes where
two people guessed correctly, so the probability of getting two people who correctly identified the percolated coffee
is 6
16 . Another way to determine the number of ways that exactly two people out of four people can identify the
percolated coffee is simply to count how many ways two people can be selected from four people:
solution on module
In addition, a graphing calculator can also be used to calculate binomial probabilities.
By pressing [2ND][DISTR], you can enter ’binompdf (4,0.5,2)’. This command calculates the binomial probability
for k (in this example, k = 2) successes out of n (in this example, n = 4) trials, when the probability of success on
any one trial is p (in this example, p = 0.5).
-read-
A binomial experiment is a probability experiment that satisfies the following conditions:
• Each trial can have only two outcomesone known as a success, and the other known as a failure.
• There must be a fixed number, n, of trials.
• The outcomes of the trials must be independent of each other. The probability of each success doesn’t change,
regardless of what occurred previously.
• The probability, p, of a success must remain the same for each trial.
The distribution of the random variable X, where x is the number of successes, is called a binomial probability
distribution. The probability that you get exactly x = k successes is as follows:
formula on module*
To apply the binomial formula to a specific problem, it is useful to have an organized strategy. Such a strategy is
presented in the following steps:
• Identify a success.
• Determine p, the probability of success.
• Determine n, the number of experiments or trials.
• Use the binomial formula to write the probability distribution of X.
Example: According to a study conducted by a telephone company, the probability is 25% that a randomly selected
phone call will last longer than the mean value of 3.8 minutes. What is the probability that out of three randomly
selected calls:
a. Exactly two last longer than 3.8 minutes?
b. None last longer than 3.8 minutes?
-answer-
Example: A car dealer knows from past experience that he can make a sale to 20% of the customers who he interacts
with. What is the probability that, in five randomly selected interactions, he will make a sale to:
a. Exactly three customers?
b. At most one customer?
c. At least one customer?
-answer-
Example: A poll of twenty voters is taken to determine the number in favor of a certain candidate for mayor. Suppose
that 60% of all the city’s voters favor this candidate.
a. Find the mean and the standard deviation of X.
b. Find the probability of x 10.
c. Find the probability of x > 12.
d. Find the probability of x = 11.
-answer-
Technology Note: Calculating Binomial Probabilities on the TI-83/84 Calculator
Use the ’binompdf(’ command to calculate the probability of exactly k successes. Press [2ND][DIST] and scroll
down to ’A:binompdf(’. Press [ENTER] to place ’binompdf(’ on your home screen. Type values of n, p, and k,
separated by commas, and press [ENTER].
Use the ’binomcdf(’ command to calculate the probability of at most x successes. The format is ’binomcdf(n, p, k)’
to find the probability that x k. (Note: It is not necessary to close the parentheses.)
-read-
A binomial experiment consists of n identical trials
-read
There are only two possible outcomes on each trial: S (for success) or F (for failure)
-read-
