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```Chapter 8
Counting Principles:
Further Probability
Topics
Section 8.4
Binomial Probability




Many probability problems are concerned with
experiments in which an event is repeated many
times.
Probability problems of this kind are called
Bernoulli trials, or Bernoulli processes.
In each case, some outcome is designated a
success, and any other outcome is considered a
failure.
Bernoulli trials problems are sometimes referred
to as binomial experiments.
Binomial Experiments
The following criteria must be met:
1.) The same experiment is repeated several times.
2.) There are only two possible outcomes, success
and failure.
3.) The repeated trials are independent, so that the
probability of success remains the same for
each trial.

A single die is rolled four times in a row.
Getting a 5 is a “success”, while getting any
other number is a “failure”.
a.) Why is this a binomial experiment?
Already classified outcomes as successes or failures.
 Experiment is repeated several times.
 Outcomes are independent when rolling a single die.


A single die is rolled four times in a row. Getting
a 5 is a “success”, while getting any other number
is a “failure”.
b.) Find the probability of having 3 successes
followed by 1 failure.
P(S  S  S  F) =
1 1 1 5
5
   
6 6 6 6 1296
A single die is rolled four times in a row. Getting a 5 is a
“success”, while getting any other number is a “failure”.
c.) Find the probability of having 3 successes
and 1 failure, in any order.
Possible Outcomes
SSSF SSFS SFSS FSSS

Thus, P(3S  1F) = 4 (
1 1 1 5
  
6 6 6 6
3
=
=
1
1 5
4   
6 6
5
 0.0154
324
)
Binomial Probability Formula

If p is the probability of success in a single trial of a
binomial experiment, the probability of x successes
and n-x failures in n independent repeated trials of the
experiment is
C
p
(1

p
)
n x
x
n = # of trials
x = # of successes
p = probability of success
n x

Example: A single die is rolled 10 times. Find the
probability of rolling exactly 7 fives.
n = # trials = 10
success: rolling a five
x = # successes = 7
p = prob. of success = 1/6
1
10 C7 

6
7
10  7
 1
1  
 6
7
3
1 5
 10 C7      2.481104
6 6

Example: A restaurant manager estimates the probability that
a newly hired waiter will still be working at the restaurant six
months later is only 60%. For the five new waiters just hired,
what is the probability that at least four of them will still be
working at the restaurant in six months? Assume that the
waiters decide independently of each other.
n=5
x = 4 or 5
p = 0.6
C4  0.6   0.4   5 C5  0.6   0.4   0.337
4
5
1
5
0

Example: A flu vaccine has a probability of 80% of
preventing a person who is inoculated from getting
the flu. A county health office inoculates 87 people.
Find the probabilities of the following.
a.) Exactly 15 of the people inoculated get the flu.
b.) No more than 3 of the people inoculated get the
flu.
c.) None of the people inoculated get the flu.
```
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