# 5.2 Mean or expected value and standard deviation  (Page 2/21)

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## Try it

A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of 50 patients, the following information was obtained. What is the expected value?

x P ( x )
0 P ( x = 0) = $\frac{4}{50}$
1 P ( x = 1) = $\frac{8}{50}$
2 P ( x = 2) = $\frac{16}{50}$
3 P ( x = 3) = $\frac{14}{50}$
4 P ( x = 4) = $\frac{6}{50}$
5 P ( x = 5) = $\frac{2}{50}$

The expected value is 2.24

(0) $\frac{4}{50}$ + (1) $\frac{4}{50}$ + (2) $\frac{16}{50}$ + (3) $\frac{14}{50}$ + (4) $\frac{6}{50}$ + (5) $\frac{2}{50}$ = 0 + $\frac{8}{50}$ + $\frac{32}{50}$ + $\frac{42}{50}$ + $\frac{24}{50}$ + $\frac{10}{50}$ = $\frac{116}{50}$ = 2.24

Suppose you play a game of chance in which five numbers are chosen from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A computer randomly selects five numbers from zero to nine with replacement. You pay $2 to play and could profit$100,000 if you match all five numbers in order (you get your $2 back plus$100,000). Over the long term, what is your expected profit of playing the game?

To do this problem, set up an expected value table for the amount of money you can profit.

Let X = the amount of money you profit. The values of x are not 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Since you are interested in your profit (or loss), the values of x are 100,000 dollars and −2 dollars.

To win, you must get all five numbers correct, in order. The probability of choosing one correct number is $\frac{1}{10}$ because there are ten numbers. You may choose a number more than once. The probability of choosing all five numbers correctly and in order is

$\left(\frac{1}{10}\right)\left(\frac{1}{10}\right)\left(\frac{1}{10}\right)\left(\frac{1}{10}\right)\left(\frac{1}{10}\right)=\left(1\right)\left({10}^{-5}\right)=0.00001.$

Therefore, the probability of winning is 0.00001 and the probability of losing is

$1-0.00001=0.99999.$

The expected value table is as follows:

x P ( x ) x * P ( x )
Loss –2 0.99999 (–2)(0.99999) = –1.99998
Profit 100,000 0.00001 (100000)(0.00001) = 1

Since –0.99998 is about –1, you would, on average, expect to lose approximately $1 for each game you play. However, each time you play, you either lose$2 or profit $100,000. The$1 is the average or expected LOSS per game after playing this game over and over.

## Try it

You are playing a game of chance in which four cards are drawn from a standard deck of 52 cards. You guess the suit of each card before it is drawn. The cards are replaced in the deck on each draw. You pay $1 to play. If you guess the right suit every time, you get your money back and$256. What is your expected profit of playing the game over the long term?

Let X = the amount of money you profit. The x -values are –$1 and$256.

The probability of guessing the right suit each time is $\left(\frac{1}{4}\right)\left(\frac{1}{4}\right)\left(\frac{1}{4}\right)\left(\frac{1}{4}\right)=\frac{1}{256}$ = 0.0039

The probability of losing is 1 – $\frac{1}{256}$ = $\frac{255}{256}$ = 0.9961

(0.0039)256 + (0.9961)(–1) = 0.9984 + (–0.9961) = 0.0023 or 0.23 cents.

Suppose you play a game with a biased coin. You play each game by tossing the coin once. P (heads) = $\frac{2}{3}$ and P (tails) = $\frac{1}{3}$ . If you toss a head, you pay $6. If you toss a tail, you win$10. If you play this game many times, will you come out ahead?

a. Define a random variable X .

a. X = amount of profit

b. Complete the following expected value table.

x ____ ____
WIN 10 $\frac{1}{3}$ ____
LOSE ____ ____ $\frac{–12}{3}$

b.

x P ( x ) xP ( x )
WIN 10 $\frac{1}{3}$ $\frac{10}{3}$
LOSE –6 $\frac{2}{3}$ $\frac{–12}{3}$

c. What is the expected value, μ ? Do you come out ahead?

c. Add the last column of the table. The expected value μ = $\frac{\text{–}2}{3}$ . You lose, on average, about 67 cents each time you play the game so you do not come out ahead.

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
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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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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