# 4.5 Geometric distribution

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

#### Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Collaborative statistics: custom version modified by r. bloom. OpenStax CNX. Nov 15, 2010 Download for free at http://legacy.cnx.org/content/col10617/1.4
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