School

Hillsborough Community College **We aren't endorsed by this school

Course

MAT 0029

Subject

Statistics

Date

Nov 20, 2023

Type

Other

Pages

6

Uploaded by ConstableFlower442 on coursehero.com

1
Section 4.2 Binomial Distributions
A.
Binomial Experiments
Definition:
A binomial experiment is a probability experiment that satisfies these
conditions.
1.
The experiment has a ______________________ number of trials, where each
trial is independent of other trials.
2.
There are only _____________ possible outcomes of interest for each trial.
Each outcome can be classified as _________________________ or
__________________________.
3.
The probability for success is ______________________________ for each trial.
4.
The random variable
࠵?
counts the number of successful trials.
Notation for Binomial Experiments
___________: the number of trials
___________: the probability of success in a single trial
___________: the probability of failure in a single trial
___________: the random variable that represents a count of the number of successes is
࠵?
trials
Note: Success __________________________ always mean that something good occurred!
Ex. 1: From a standard deck of cards, you pick a card, note whether or not the card is
a club, and then replace the card. This experiment is repeated 5 times. Does this
represent a binomial experiment? If so, find
࠵?, ࠵?, ࠵?, and list all possible values of ࠵?.
If
it is not a binomial experiment, explain why.
fixed
2
a
success
a
failure
the
same
n
p
9
i
p
g
does
not
Binomial
Clubs
or
Not
Clubs
n
S
p
132
4
25
9
1
p
1
25
075
x
of
trials
successful
x
0
I
2,3
4,5

2
B.
Binomial Probability Formula
In a binomial experiment, the probability of exactly
࠵?
successes in
࠵?
trials is given by
the formula
࠵?(࠵?) = ࠵?(࠵?, ࠵?)࠵?
௫
࠵?
ି௫
=
࠵?!
(࠵? − ࠵?)! ࠵?!
࠵?
௫
࠵?
ି௫
.
The number of failures is
࠵? − ࠵?.
Ex. 2: Rotator cuff surgery has a 90% chance of success. The surgery is performed
on three patients. Find the probability of the surgery being successful on exactly two
patients.
C.
Mean, Variance, and Standard Deviations of Binomial Probability
Distributions
By listing the possible values of
࠵?
with the corresponding probabilities, we can
construct a binomial probability distribution.
If
࠵?
is a binomial random variable,
࠵?
is the total number of trials,
࠵?
is the probability
of success and
࠵?
is the probability of failure, then
Expected Value:
࠵?(࠵?) = ࠵? = ࠵?࠵?
Variance:
࠵?(࠵?) = ࠵?
ଶ
= ࠵?࠵?࠵?
Standard Deviation:
࠵?࠵?(࠵?) = ࠵? =
ඥ
࠵?࠵?࠵?
nCxpxqn
X
p
90
g
10
n
3
x
2
P
x
R
Cx
p
g
n
x
PIX
302
x
1.907
x
C
10
0.2430
F
S
F
S
Not

3
Ex. 3: Thirty-nine percent of U.S. adults have very little or no confidence in
newspapers. You randomly select eight U.S. adults. Find the probability that the
number who have very little or no confidence in the newspapers is
a.
Exactly six.
b.
Exactly three.
Ex. 4: The probability of success in an experiment is 0.65. If the experiment is
conducted 10 times, what is the probability of at least six successes?
Ex. 5: Fourteen percent of consumers have tried to purchase clothing second-hand
rather than new in the past year. You randomly select 11 consumers. Find the
probability that the number who have tried to purchase second-hand rather than
new clothing is
a.
At most five
b.
More than three
p
39
9
1
39
61
x
6
n
8
PIG
Cake
39
6172
0367
or
037
x
3
P
3
g
Cz
x
1.3973
x
C
6175
0.281
p
0.65
9
1
p
35
n
10
X
Z
6
X
26
P
6
P 7
PC8
PIG
P
IO
10667
1.65767
61.3574
10
C
C
65771.3573
to
Ca
1
65781.355
local
65791.35
t
10C
to
C
657101.35
0
2377
0.2522
0.1757
0.0725
0.0135
I
0.7516
0
n
11
p
14
9
II
is
s
PIX
I
5
PCO
PCI
P
12
P
3
P
14
PCS
p
Col
14701.86
t
1
C
C
14711.86310
t
C2
C
14721.8679
n
Cz
14731.867
1
Cy
l
14741.8677
n
Cs C
74751.8636
1903
t
3408
t
2774
1
355
t
0441
t
0101
I
99820
x
7
3
PIX
3
P
4
P
15
P
6
P1
P
8
TP
9
P
IO
P
Il
FX
3
1
P
x
3
1
Plo
PCI
P 2
P
133
1
in
Col
14701.86
ti
C
C
14
86110
1
C2
C
14721.8679
1636.143686