CHS Statistics
Objective: To solve multistep probability
tasks with the concept of binomial
distribution
Warm-Up

 AT&T directory assistance provides customers with
correct responses 90% of the time. Using your
method of choice, find the following probability:
 P(exactly 3 correct responses for 5 directory assistance
requests)
Binomial Distributions

Bernoulli Trials
 A binomial distribution will have the following
requirements:
1. The procedure has a fixed number of trials.
2. The trials must be independent (the outcome of any
individual trial does not affect the probabilities in the
other trials).
3. Each trial must have all outcomes classified into two
categories (usually success or failure).
4. The probabilities must remain constant for each trial.
Notation for Binomial Distributions

n = the total fixed number of trials
x = the specific number of successes in n trials
p = the probability of success
q = the probability of failure (1 – p)
Examples

 You select a card from a standard deck. You look to
see if it is a club, then replace the card. You do this
experiment 5 times. Find n, p, q, and possible values
of x.
Examples (cont.)

 Decide whether the following experiments are binomial
experiments. If they are, specify the values of n, p, q, and
list possible values of x. If not, explain.
 A certain surgical procedure has an 85% chance of success. A
doctor performs the procedure on 8 patients. The random
variable represents the number of successful surgeries.
 A jar contains five red marbles, nine blue marbles, and six
green marbles. You randomly select three marbles from the jar,
without replacement. The random variable represents the
number of red marbles.
Examples (cont.)

 Decide whether the following experiments are binomial
experiments. If they are, specify the values of n, p, q, and
list possible values of x. If not, explain.
 You take a multiple choice quiz that consists of 10 questions.
Each question has four possible answers, one of which is
correct. To complete the quiz, you randomly guess the answer
to each question. The random variable represents the number
of correct answers.
Binomial Probability Formula

 The probability of exactly x successes in n trials is:
P(X) = nCx ∙ 𝒑𝒙 ∙ 𝒒𝒏−𝒙
or
P(X) =
𝒏
𝒙
∙ 𝒑𝒙 ∙ 𝒒𝒏−𝒙
n = the total fixed number of trials
x = the specific number of successes in n trials
p = the probability of success
q = the probability of failure (1 – p)
Example

Micro-fracture knee surgery has a 75% chance of success on
patients with degenerative knees. The surgery is performed
on three patients.
 Find the probability of the surgery being successful on
exactly two patients.
 Find the probability that fewer than two are successful.
Binomial Probability
Calculator Techniques

 2nd  DISTR  binompdf(
 Note the pdf for Probability Density Function
 Used to find any individual outcome
 Format: binompdf(n,p,x)
 2nd  DISTR  binomcdf(
 Note the cdf for Cumulative Density Function
 Used for getting x or fewer successes among n trials
 Format: binomcdf(n,p,x)
 Note: if you wanted to find up to a #, use the complement rule.
All possible probabilities in the model will add up to 1.
Examples

Let’s complete the warm-up using binomial distribution
formulas:
 Find the probability of getting exactly 3 correct responses
among 5 different requests from At&T directory
assistance. Assume that in general, AT&T is correct 90%
of the time.
 Find the probability of less than 2
 Find the probability of more than 1.
Examples (cont.)

A survey indicates that 41% of women in the US consider reading
their favorite leisure time activity. You randomly select four US
women and ask them if reading is their favorite leisure-time
activity.
 Find the probability that exactly two of them respond that reading
is their favorite leisure time activity.
 Find the probability that at least 3 respond that reading is their
favorite leisure time activity.
 Find the probability that fewer than 1 respond that reading is their
favorite leisure time activity.
Examples (cont.)

A survey indicates that 21% of men in the US consider fishing their
favorite leisure-time activity. You randomly select five US men and ask
them whether fishing is their favorite leisure-time activity.
 Find the probability that exactly three of them respond that fishing is
their favorite leisure-time activity.
 Find the probability that at least 4 respond that fishing is their favorite
leisure-time activity.
 Find the probability that more than 1 respond that fishing is their
favorite leisure-time activity.
4.4
Mean and Standard Deviation
of a Binomial Distribution

Mean: 𝜇 = 𝑛𝑝
Standard Deviation: 𝜎 = 𝑛𝑝𝑞
Example

 In Pittsburgh, about 56% of the days in a year are
cloudy. Find the mean and standard deviation for
the number of cloudy days in the month of June.
Assignment

 Pp. 401-404 # 1, 3, 7, 17 (skip a), 19 (skip c), 21
 Be sure to check your answers with the solutions
manual online.