# 4.5 Geometric (optional)

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This module describes the geometric experiment and the geometric probability distribution. This module is included in the Collaborative Statistics textbook/collection as an optional lesson.

The characteristics of a geometric experiment are:

1. There are one or more Bernoulli trials with all failures except the last one, which is a success. In other words, you keep repeating what you are doing until the first success.Then you stop. For example, you throw a dart at a bull's eye until you hit the bull's eye. The first time you hit the bull's eye is a "success" so you stop throwing the dart. It mighttake you 6 tries until you hit the bull's eye. You can think of the trials as failure, failure, failure, failure, failure, success. STOP.
2. In theory, the number of trials could go on forever. There must be at least one trial.
3. The probability, $p$ , of a success and the probability, $q$ , of a failure is the same for each trial. $(p+q, 1)$ and $(q, 1-p)$ . For example, the probability of rolling a 3 when youthrow one fair die is $\frac{1}{6}$ . This is true no matter how many times you roll the die. Suppose you want to know the probability of getting the first 3 on the fifth roll. On rolls1, 2, 3, and 4, you do not get a face with a 3. The probability for each of rolls 1, 2, 3, and 4 is $(q, \frac{5}{6})$ , the probability of a failure. The probability of getting a 3 on the fifthroll is $(\frac{5}{6}\cdot \frac{5}{6}\cdot \frac{5}{6}\cdot \frac{5}{6}\cdot \frac{1}{6}, 0.0804)$
$X=$ the number of independent trials until the first success. The mean and variance are in the summary in this chapter.

You play a game of chance that you can either win or lose (there are no other possibilities) until you lose. Your probability of losing is $\mathrm{p = 0.57}$ . What is the probability that it takes 5 games until you lose? Let $X$ = the number of games you play until you lose (includes the losing game). Then $X$ takes on the values 1, 2, 3, ... (could go on indefinitely). The probability question is $(P\left(x, 5\right))$ .

A safety engineer feels that 35% of all industrial accidents in her plant are caused by failure of employees to follow instructions. She decides to look at theaccident reports (selected randomly and replaced in the pile after reading) until she finds one that shows an accident caused by failure of employees to follow instructions. On the average, how many reports would the safetyengineer expect to look at until she finds a report showing an accident caused by employee failure to follow instructions? What is the probability that the safety engineerwill have to examine at least 3 reports until she finds a report showing an accident caused by employee failure to follow instructions?

Let $X$ = the number of accidents the safety engineer must examine until she finds a report showing an accident caused by employee failure to follow instructions. $X$ takes on the values 1, 2, 3, .... The first question asks you to find the expected value or the mean. The second question asks you to find $(P\left(x, 3\right))$ . ("At least" translates as a"greater than or equal to" symbol).

Suppose that you are looking for a student at your college who lives within five miles of you. You know that 55% of the 25,000 students do live within five miles of you. You randomly contact students from the college until one says he/she lives within five miles of you. What is the probability that you need to contact four people?

This is a geometric problem because you may have a number of failures before you have the one success you desire. Also, the probability of a success stays the same each time you ask astudent if he/she lives within five miles of you. There is no definite number of trials (number of times you ask a student).

Let $X$ = the number of ____________ you must ask ____________ one says yes.

Let $X$ = the number of students you must ask until one says yes.

What values does $X$ take on?

1, 2, 3, …, (total number of students)

What are $p$ and $q$ ?

• $p$ = 0.55
• $q$ = 0.45

The probability question is P(_______).

## Notation for the geometric: g = geometric probability distribution function

$X$ ~ $\mathrm{G\left(p\right)}$

Read this as " $X$ is a random variable with a geometric distribution." The parameter is $p$ . $p$ = the probability of a success for each trial.

Assume that the probability of a defective computer component is 0.02. Components are randomly selected. Find the probability that the first defect is caused by the 7th componenttested. How many components do you expect to test until one is found to be defective?

Let $X$ = the number of computer components tested until the first defect is found.

$X$ takes on the values 1, 2, 3, ... where $p=0.02$ . $X$ ~ $\text{G(0.02)}$

Find $(P\left(x, 7\right))$ . $((P\left(x, 7\right)), 0.0177)$ . (calculator or computer)

TI-83+ and TI-84: For a general discussion, see this example (binomial) . The syntax is similar. The geometric parameter list is (p, number) If "number" is left out, the result is thegeometric probability table. For this problem: After you are in 2nd DISTR, arrow down to D:geometpdf. Press ENTER. Enter .02,7). The result is $((P\left(x, 7\right)), 0.0177)$ .

The probability that the 7th component is the first defect is 0.0177.

The graph of $X$ ~ $\text{G(0.02)}$ is:

The $y$ -axis contains the probability of $x$ , where $X$ = the number of computer components tested.

The number of components that you would expect to test until you find the first defective one is the mean, $\mu$ = 50.

The formula for the mean is $(((\mu , \frac{1}{p}), \frac{1}{0.02}), 50)$

The formula for the variance is $((({\sigma }^{2}, \frac{1}{p}\cdot \left(\frac{1}{p}-1\right)), \frac{1}{0.02}\cdot \left(\frac{1}{0.02}-1\right)), 2450)$

The standard deviation is $(((\sigma , \sqrt{\frac{1}{p}\cdot \left(\frac{1}{p}-1\right)}), \sqrt{\frac{1}{0.02}\cdot \left(\frac{1}{0.02}-1\right)}), 49.5)$

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.