# 5.3 Binomial distribution - university of calgary - base content  (Page 2/30)

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A fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than ten heads? Let X = the number of heads in 15 flips of the fair coin. X takes on the values 0, 1, 2, 3, ..., 15. Since the coin is fair, p = 0.5 and q = 0.5. The number of trials is n = 15. State the probability question mathematically.

P ( X >10)

## Try it

A fair, six-sided die is rolled ten times. Each roll is independent. You want to find the probability of rolling a one more than three times. State the probability question mathematically.

P ( X >3)

Approximately 70% of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time? Students are selected randomly.

a. This is a binomial problem because there is only a success or a __________, there are a fixed number of trials, and the probability of a success is 0.70 for each trial.

a. failure

b. If we are interested in the number of students who do their homework on time, then how do we define X ?

b. X = the number of statistics students who do their homework on time

c. What values does x take on?

c. 0, 1, 2, …, 50

d. What is a "failure," in words?

d. Failure is defined as a student who does not complete his or her homework on time.

The probability of a success is p = 0.70. The number of trials is n = 50.

e. If p + q = 1, then what is q ?

e. q = 0.30

f. The words "at least" translate as what kind of inequality for the probability question P ( X ____ 40).

f. greater than or equal to (≥)
The probability question is P ( X ≥ 40).

## Try it

Sixty-five percent of people pass the state driver’s exam on the first try. A group of 50 individuals who have taken the driver’s exam is randomly selected. Give two reasons why this is a binomial problem.

This is a binomial problem because there is only a success or a failure, and there are a definite number of trials. The probability of a success stays the same for each trial.

## Notation for the binomial: B = binomial probability distribution function

X ~ B ( n , p )

Read this as " X is a random variable with a binomial distribution." The parameters are n and p ; n = number of trials, p = probability of a success on each trial.

## Binomial formula

P ( X = x )= $\left(\begin{array}{c}n\\ x\end{array}\right)p^{x}q^{\mathrm{n-x}}$

• $p^{x}$ is the probability of x successes in x independent and identical trials. For example, if the probability of success = 0.4, the probability of five successes in five independent and identical trials is
$0.4\times 0.4\times 0.4\times 0.4\times 0.4=\mathrm{0.4}^{5}=p^{x}$
• $q^{\mathrm{n-x}}$ is the probability of n - x failures in n - x identical and independent trials. For example, if the probability of success = 0.4, then the probability of failure is 1 - p = 0.6. If there are eight trials ( n = 8) with five successes ( x = 5 ), then there were three failures in the eight trials ( n - x = 8 - 5 = 3). The probability of three failures in three independent and identical trials is
$0.6\times 0.6\times 0.6=\mathrm{0.6}^{3}=q^{\mathrm{n-x}}$
• $\left(\begin{array}{c}n\\ x\end{array}\right)$ represents the number of combinations of x successes in n trials. If there are eight trials ( n = 8) and five successes ( x = 5), then there are 56 possible ways to arrange five successes among eight trials.
$\left(\begin{array}{c}8\\ 5\end{array}\right)=\mathrm{56}$
• The formula P ( X = x )= $\left(\begin{array}{c}n\\ x\end{array}\right)p^{x}q^{\mathrm{n-x}}$ is the probability of x successes in n independent and identical trials. If there are eight independent and identical trials, the probability of five successes where p = 0.4 is
$P(\mathrm{X=5})=\left(\begin{array}{c}8\\ 5\end{array}\right)\mathrm{0.4}^{5}\mathrm{0.6}^{3}=\mathrm{0.1239}$

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
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Stoney Reply
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?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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characteristics of micro business
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Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Introduction to statistics i - stat 213 - university of calgary - ver2015revb. OpenStax CNX. Oct 21, 2015 Download for free at http://legacy.cnx.org/content/col11874/1.3
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