This document provides information about the binomial probability distribution, including its basics and terminology. A binomial experiment has n repeated trials with two possible outcomes (success/failure), where the probability of success p remains constant from trial to trial. The number of successes X in n trials is a binomial random variable with a binomial probability distribution given by the formula P(X)=Cn*px*q^n-x, where q is the probability of failure. Several examples demonstrate calculating probabilities of outcomes for binomial experiments.
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Chapter 5: Discrete Probability Distribution
5.2 - Binomial Probability Distributions
Solution to the Practice Test 3A, Chapter 6 Normal Probability DistributionLong Beach City College
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Elementary Statistics Practice Test 3
Practice Test Chapter 6 (Normal Probability Distributions)
Chapter 6: Normal Probability Distributions
The following presentation is an introduction to the Algebraic Methods – part one for level 4 Mathematics. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
Detail Description about Probability Distribution for Dummies. The contents are about random variables, its types(Discrete and Continuous) , it's distribution (Discrete probability distribution and probability density function), Expected value, Binomial, Poisson and Normal Distribution usage and solved example for each topic.
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Chapter 5: Discrete Probability Distribution
5.2 - Binomial Probability Distributions
Solution to the Practice Test 3A, Chapter 6 Normal Probability DistributionLong Beach City College
Please Subscribe to this Channel for more solutions and lectures
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Elementary Statistics Practice Test 3
Practice Test Chapter 6 (Normal Probability Distributions)
Chapter 6: Normal Probability Distributions
The following presentation is an introduction to the Algebraic Methods – part one for level 4 Mathematics. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
Detail Description about Probability Distribution for Dummies. The contents are about random variables, its types(Discrete and Continuous) , it's distribution (Discrete probability distribution and probability density function), Expected value, Binomial, Poisson and Normal Distribution usage and solved example for each topic.
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2. BASICS & TERMINOLOGY:
> Outcome:- The end result of an experiment.
> Random experiment
predictable.
> Random Event:- A
Experiments whose outcomes are not
random event is an outcome or set of
outcomes of a random experiment that share a common attribute.
> Sample space:- The sample space is an exhaustive list of all the
possible outcomes of an experiment, which is usually denoted by S
3. > Random Variables.
□ Discrete Random Variable .
□ Continuous Random Variable.
> Bernoulli Trial:
A trial having only two possible outcomes is called
Bernoulli trials.
example:
□ success or failure
□ head or tail
4. > Binomial Distribution:-
The Binomial Distribution describes discrete , not
continuous, data, resulting from an experiment known as
Bernoulli process.
5. A binomial experiment is one that possesses
the following properties:
► The experiment consists of n repeated trials;
► Each trial results in an outcome that may be classified as a success or a failure
(hence the name, binomial);
► The probability of a success, denoted by p, remains constant from trial to trial and
repeated trials are independent.
► The number of successes X in n trials of a binomial experiment is called a
binomial random variable.
► The probability distribution of the random variable X is called a binomial
distribution, and is given by the formula:
P(X) = Cn
xpxqn-x
6. EXAMPLE 1:
► Hospital records show that of patients suffering from a
certain disease, 75% die of it. What is the probability that
of 6 randomly selected patients, 4 will recover?
7. This is a binomial distribution because there are only 2
outcomes (the patient dies, or does not).
Let X = number who recover.
Here, // = 6 and ,v = 4. Letp = 0.25 (success - i.e. they live), <y
= 0.75 (failure, i.e. they die).
The probability that 4 will recover:
PLY) =
= C6
a(
0.25)4(0.75)2 =
15x 2.1973
xlO“3
0.0329595
8. EXAMPLE 3:
In the old days, there was a probability of 0.8 of success in
any attempt to make a telephone call.
(This often depended on the importance of the person
making the call, or the operator's curiosity!)
Calculate the probability of having 7 successes in 10
attempts.
9. Probability of success p = 0.8,
so q- 0.2. X=success in
getting through.
Probability of 7 successes in
10 attempts:
Probability = P(X = 7)
= Cj°(0.8)7 (0.2)10-7 = 0.20133
10.
11. EXAMPLE 4:
The ratio of boys to girls at birth in Singapore is quite high at
1.09:1.
What proportion of Singapore families with exactly 6 children
will have at least 3 boys?
(Ignore the probability of multiple births.)
[Interesting and disturbing trivia: In most countries the
ratio of boys to girls is about 1.04:1, but in China it is
1.15:1.]
12. he probability of getting a boy is j QQ — 0. 521 5
et X = number of boys in the family
ere,
n = 6 ,
p = 0 5215,
<7 = 1 - 0 52153 = 0 4785
- 3 P(X) - Cn
xp*q- C|(0.5215)3(0.4785)3 - 0.31077
- 4 P(X) - C5(0.5215)4(0.4785)2 - 0.25402
- 5 P(X) - Cf(0.5215)5(0.4785)1 - 0. 1 1074
- 6 P(X) - C|(0.5215)6 (0.4785)° - 2. 0 1 1 5 x 10'2
o the probability of getting at least 3 boys is:
Probability = P(X >3)
- 0.31077+0.25402 + 0. 11074 + 2. Oil 5x 102
- 0.69565
NOTE: We could have calculated it like this:
P{X > 3) - 1 - (PC*0) + P(*,) + P(*2))
13. EXAMPLE 5:
A manufacturer of metal pistons finds that on the average, 12% of his
pistons are rejected because they are either oversize or undersize. What
is the probability that a batch of 10 pistons will contain
(a) no more than 2 rejects?
(b) (b) at least 2 rejects?
14. Let X = number of rejected pistons (In this case,
"success” means rejection!)
Here, n = 10, p = 0.12, q = 0.88.
(a)
No rejects
x - 0 P(X) - C”p*q*~x - cj°(0. 12)0(0.88)'°
One reject
AT - 1 P(X) - C|°(0. 12)'(0.88)® - 0.37977 Two rejects
x - 2 P(X) - Cj°(0. I2)2(0. 88)S - 0.23304 So the probability
of getting no more than 2 rejects is:
Probability = P(X < 2)
= 0.278 5+ 0.3 797 7 4-0.233 04
16. BINOMIAL FREQUENCY DISTRIBUTION
► If the binomial probability distribution is multiplied by
the number of experiment N then the distribution is
called binomial frequency distribution.
f(X=x)= N P(X=x)=N
v.
n
x
qn x px
17. MEAN & VARIANCE:
Mean, p = n*p
Std. Dev. s = V ” * P
Variance, s2 =n*p*q
Where
n = number of fixed trials p = probability of
success in one of the n trials q = probability of
failure in one of the n trials
18. Applications for binomial distributions
> Binomial distributions describe the possible number of times that a
particular event will occur in a sequence of observations.
> They are used when we want to know about the occurrence of an
event, not its magnitude.
Example:
> In a clinical trial, a patient’s condition may improve or not. We study
the number of patients who improved, not how much better they feel.
> Is a person ambitious or not? The binomial distribution describes the
number of ambitious persons, not how ambitious they are.
> In quality control we assess the number of defective items in a lot
^^iaaQds,irrespective of the type of defect.