# 15.2 Problems on random selection  (Page 3/7)

A game is played as follows:

1. A wheel is spun, giving one of the integers 0 through 9 on an equally likely basis.
2. A single die is thrown the number of times indicated by the result of the spin of the wheel. The number of points made is the total of the numbers turned up on the sequence ofthrows of the die.
3. A player pays sixteen dollars to play; a dollar is returned for each point made.

Let Y represent the number of points made and $X=Y-16$ be the net gain (possibly negative) of the player. Determine the maximum value of

X , $E\left[X\right]$ , $\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]$ , $P\left(X>0\right)$ , $P\left(X>=10\right)$ , $P\left(X>=16\right)$ .

gn = 0.1*ones(1,10); gy = (1/6)*[0 ones(1,6)]; [Y,PY]= gendf(gn,gy); [X,PX]= csort(Y-16,PY); M = max(X)M = 38 EX = dot(X,PX) % Check EX = En*Ey - 16 = 4.5*3.5EX = -0.2500 % 4.5*3.5 - 16 = -0.25 VX = dot(X.^2,PX) - EX^2VX = 114.1875 Ppos = (X>0)*PX' Ppos = 0.4667P10 = (X>=10)*PX' P10 = 0.2147P16 = (X>=16)*PX' P16 = 0.0803

Marvin calls on four customers. With probability ${p}_{1}=0.6$ he makes a sale in each case. Geraldine calls on five customers, with probability ${p}_{2}=0.5$ of a sale in each case. Customers who buy do so on an iid basis, and order an amount Y i (in dollars) with common distribution:

$Y=\left[200\phantom{\rule{0.277778em}{0ex}}220\phantom{\rule{0.277778em}{0ex}}240\phantom{\rule{0.277778em}{0ex}}260\phantom{\rule{0.277778em}{0ex}}280\phantom{\rule{0.277778em}{0ex}}300\right]\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}PY=\left[0.10\phantom{\rule{0.277778em}{0ex}}0.15\phantom{\rule{0.277778em}{0ex}}0.25\phantom{\rule{0.277778em}{0ex}}0.25\phantom{\rule{0.277778em}{0ex}}0.15\phantom{\rule{0.277778em}{0ex}}0.10\right]$

Let D 1 be the total sales for Marvin and D 2 the total sales for Geraldine. Let $D={D}_{1}+{D}_{2}$ . Determine the distribution and mean and variance for D 1 , D 2 , and D . Determine $P\left({D}_{1}\ge {D}_{2}\right)$ and $P\left(D\ge 1500\right)$ , $P\left(D\ge 1000\right)$ , and $P\left(D\ge 750\right)$ .

gnM = ibinom(4,0.6,0:4); gnG = ibinom(5,0.5,0:5);Y = 200:20:300; PY = 0.01*[10 15 25 25 15 10]; [D1,PD1]= mgdf(gnM,Y,PY); [D2,PD2]= mgdf(gnG,Y,PY); ED1 = dot(D1,PD1)ED1 = 600.0000 % Check: ED1 = EnM*EY = 2.4*250 VD1 = dot(D1.^2,PD1) - ED1^2VD1 = 6.1968e+04 ED2 = dot(D2,PD2)ED2 = 625.0000 % Check: ED2 = EnG*EY = 2.5*250 VD2 = dot(D2.^2,PD2) - ED2^2VD2 = 8.0175e+04 [D1,D2,t,u,PD1,PD2,P]= icalcf(D1,D2,PD1,PD2); Use array opertions on matrices X, Y, PX, PY, t, u, and P[D,PD] = csort(t+u,P);ED = dot(D,PD) ED = 1.2250e+03eD = ED1 + ED2 % Check: ED = ED1 + ED2 eD = 1.2250e+03 % (Continued next page)VD = dot(D.^2,PD) - ED^2VD = 1.4214e+05 vD = VD1 + VD2 % Check: VD = VD1 + VD2vD = 1.4214e+05 P1g2 = total((t>u).*P) P1g2 = 0.4612k = [1500 1000 750];PDk = zeros(1,3); for i = 1:3PDk(i) = (D>=k(i))*PD'; enddisp(PDk) 0.2556 0.7326 0.8872

A questionnaire is sent to twenty persons. The number who reply is a random number $N\sim$ binomial (20, 0.7). If each respondent has probability $p=0.8$ of favoring a certain proposition, what is the probability of ten or more favorable replies? Of fifteen or more?

gN = ibinom(20,0.7,0:20); gY = [0.2 0.8]; gendDo not forget zero coefficients for missing powers Enter gen fn COEFFICIENTS for gN gNEnter gen fn COEFFICIENTS for gY gY Results are in N, PN, Y, PY, D, PD, PMay use jcalc or jcalcf on N, D, P To view the distribution, call for gD.P10 = (D>=10)*PD' P10 = 0.7788P15 = (D>=15)*PD' P15 = 0.0660pD = ibinom(20,0.7*0.8,0:20); % Alternate: use D binomial (pp0) D = 0:20;p10 = (D>=10)*pD' p10 = 0.7788p15 = (D>=15)*pD' p15 = 0.0660

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
for teaching engĺish at school how nano technology help us
Anassong
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
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.
what is nano technology
what is system testing?
preparation of nanomaterial
how did you get the value of 2000N.What calculations are needed to arrive at it
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive