College Prep Stats Section 5.3 Notes Binomial Probability

College Prep Stats
Section 5.3 Notes
Binomial Probability Distribution
A binomial probability distribution results from a procedure that meets all the following requirements:
1. The procedure has a fixed number of trials.
2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.)
{Remember:
3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure).
4. The probability of a success remains the same in all trials.
Notation for Binomial Probability Distributions
S and F (success and failure) denote the two possible categories of all outcomes; p and q will denote the probabilities of S and F,
respectively, so
P(S) = p (p = probability of success)
P(F) = 1 – p = q (q = probability of failure)
n
x
p
q
P(x)
denotes the fixed number of trials.
denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive.
denotes the probability of success in one of the n trials.
denotes the probability of failure in one of the n trials.
denotes the probability of getting exactly x successes among the n trials.
Example 1: Use the following scenario to answer the questions: Consider treating 863 subjects with Lipitor (Atorvastatin) and
recording whether there is a “yes” response when they are asked if they experienced a headache (based on data from Pfizer, Inc.).
Suppose we want to find the probability that 571 said “yes”.
a) Determine whether or not the given procedure results in a binomial distribution. If it is not binomial, identify at least one
requirement that is not satisfied.
b) If it is binomial, identify the values of n, x, p, and q.
Example 2: Use the following scenario to answer the questions: Consider treating 152 couples with YSORT gender selection method
developed by the Genetics & IVF Institute and recording the ages of the parents. Suppose we want to find the probability of choosing
a parent the age of 30.
a) Determine whether or not the given procedure results in a binomial distribution. If it is not binomial, identify at least one
requirement that is not satisfied.
b) If it is binomial, identify the values of n, x, p, and q.
Example 3: Twenty different Senators are randomly selected from the 100 Senators in current Congress, and each was asked whether
he or she is in favor of abolishing state taxes. Suppose we want to find the probability that 15 are in favor of abolishing state taxes.
a) Determine whether or not the given procedure results in a binomial distribution. If it is not binomial, identify at least one
requirement that is not satisfied.
b) If it is binomial, identify the values of n, x, p, and q.
Important Hints
 Be sure that x and p both refer to the same category being called a success.
 When sampling without replacement, consider events to be independent if n < 0.05N.
Methods for Finding Probabilities
We will now discuss four methods for finding the probabilities corresponding to the random variable x in a binomial distribution.
Method 1: Using the Binomial Probability Formula =
P ( x)
n!
• p x • q n − x for x = 0, 1, 2, . . ., n
( n − x )! x !
where
n = number of trials
x = number of successes among n trials
p = probability of success in any one trial
q = probability of failure in any one trial (q = 1 – p)
Binomial Probability Formula:
P ( x)
=
n!
• p x • qn− x
( n − x )! x !
The probability of x successes among n trials
for any one particular order
Number of outcomes with exactly x
successes among n trials
Example 4: Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability
formula to find the probability of x successes given the probability p of success on a single trial.
a) n = 12, x = 10, p = 3/4
b) n = 20, x = 4, p = 0.15
Method 2: Using Minitab Table Technology
Minitab Table results can be used to find binomial probabilities.
Minitab
Example 5: Refer to the Minitab display. (When blood donors were randomly selected, 45% of them had blood that is Group O
(based on data from the Greater New York Blood Program).)
x P(x)
The display shows the probabilities obtained by entering the values of n = 5 and p = 0.45.
a) Find the probability that at least 1 of the 5 donors has Group O blood.
b) If at least 1 Group O donor is needed, is it reasonable to expect that at least 1will be obtained?
0
0.050328
1
0.205889
2
0.336909
3
0.275653
4
0.112767
5
0.018453
Method 3: Using TI-83/84 Technology
We will be doing binomial probabilities using the calculator:
Keyword “exact”
…find probability with binompdf(number of trials, probability of success, number of successes)
Keywords “less than or equal to”
…find probability with binomcdf(number of trials, probability of success, number of successes)
Keywords “at least” or “greater than or equal to” …. use 1 – binomcdf with 1 less number for success
What you are doing here is 1 minus what you DO NOT WANT and that will give you the probability that you do want.
Example 6: Use the TI 83/84 calculator to calculate the following probabilities.
a) Calculate the probability of tossing a coin 20 times and getting exactly 9 heads.
b) Calculate the probability of tossing a coin 20 times and getting less than 6 heads.
c) Calculate the probability of tossing a coin 32 times and getting at least 14 heads.
d) About 1% of people are allergic to bee stings. What is the probability that exactly 1 person in a class of 25 is allergic to bee stings?
e) Refer to (d), what is the probability that 4 or more of them are allergic to bee stings?
Example 7: Use the TI 83/84 calculator to calculate the following probabilities.
a) At a college, 53% of students receive financial aid. In a random group of 9 students, what is the probability that exactly 5 of them
receive financial aid?
b) Now using (a), what is the probability that fewer than 3 students in the class receive financial aid?
Method 4: Using Table A-1 in Appendix A
Part of Table A-1 is shown below. With n = 12 and p = 0.80 in the binomial distribution, the probabilities of 4, 5, 6, and 7 successes
are 0.001, 0.003, 0.016, and 0.053 respectively.
Example 8: Assume that a procedure yields a binomial distribution with a trial repeated n times. Use Table A-1 to find the probability
of x successes given the probability p of success on a given trial.
a) n = 2, x = 1, p = 0.3
b) n = 15, x = 11, p = 0.99
c) n = 10, x = 2, p = 0.05
Example 9: The brand name of Mrs. Fields (cookies) has a 90% recognition rate (based on data from Franchise Advantage). If Mrs.
Fields herself wants to verify that rate by beginning with a small sample of 10 randomly selected consumers, find the probability that
exactly 9 of the 10 consumers recognize her brand name. Also find the probability that the number who recognize her brand name is
not 9.

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