# A gambling book recommends the following "winning strategy" for the game of roulette. It recommends that a gambler bet $1 onred. If red appears (which has probablity 18/38), then the gamblershould take her$1 profit and quit. If the gambler loses this bet (which has probablity 20/38 of occurring), she should make additional $1 bets on red on each of the next two spins of the roulette wheel and then quite. Let X denote the gambler's winnings when she quites. (a) Find P{X > 0}. (b) Are you concinved that the strategy is indeed a "winning" strategy? Explain your answer. (c) Find E[X]. # A gambling book recommends the following "winning strategy" for the game of roulette. It recommends that a gambler bet$1 onred. If red appears (which has probablity 18/38), then the gamblershould take her $1 profit and quit. If the gambler loses this bet (which has probablity 20/38 of occurring), she should make additional$1 bets on red on each of the next two spins of the roulette wheel and then quite. Let X denote the gambler's winnings when she quites. (a) Find P{X > 0}. (b) Are you concinved that the strategy is indeed a "winning" strategy? Explain your answer. (c) Find E[X].

Question
A gambling book recommends the following "winning strategy" for the game of roulette. It recommends that a gambler bet $1 onred. If red appears (which has probablity 18/38), then the gamblershould take her$1 profit and quit. If the gambler loses this bet (which has probablity 20/38 of occurring), she should make additional $1 bets on red on each of the next two spins of the roulette wheel and then quite. Let X denote the gambler's winnings when she quites. (a) Find P{X > 0}. (b) Are you concinved that the strategy is indeed a "winning" strategy? Explain your answer. (c) Find E[X]. ## Answers (1) 2020-12-07 a) Remember that when you bet$1 and lose the bet,you lose the $1. Therefore, you have to earn$2 just to walk out with a net gain of \$1. So the answer is:
$$\displaystyle{P}{\left\lbrace{X}{>}{0}\right\rbrace}={\left({\frac{{{18}}}{{{38}}}}\right)}+{\left({\left({\frac{{{20}}}{{{38}}}}\right)}\cdot{\left({\frac{{{18}}}{{{38}}}}\right)}\cdot{\left({\frac{{{18}}}{{{38}}}}\right)}\right)}={.5918}$$
b) To get this answer, you really need to do part first. I'll come back to this after that.
c) Remember that there is a chance that you can lose some dollars when playing. So the answer is as follows:
$$\displaystyle{E}{\left[{X}\right]}=-{3}{\left[{P}{\left(-{3}\right)}\right]}+-{2}{\left[{P}{\left(-{2}\right)}\right]}+-{1}{\left[{P}{\left(-{1}\right)}\right]}+{0}{\left[{P}{\left({0}\right)}\right]}+{1}{\left[{P}{\left({1}\right)}\right]}$$
$$\displaystyle=-{3}{\left({\left({\frac{{{20}}}{{{38}}}}\right)}\cdot{\left({\frac{{{20}}}{{{38}}}}\right)}\cdot{\left({\frac{{{20}}}{{{38}}}}\right)}\right)}+{\left(-{2}\right)}{\left({0}\right)}+{\left(-{1}\right)}{\left[{\left({\left({\frac{{{20}}}{{{38}}}}\right)}\cdot{\left({\frac{{{18}}}{{{38}}}}\right)}\cdot{\left({\frac{{{20}}}{{{38}}}}\right)}\right)}+{\left({\left({\frac{{{20}}}{{{38}}}}\right)}\cdot{\left({\frac{{{20}}}{{{38}}}}\right)}\cdot{\left({\frac{{{18}}}{{{38}}}}\right)}\right)}\right]}+{0}{\left({0}\right)}+{1}{\left[{\left({\left({\frac{{{20}}}{{{38}}}}\right)}\cdot{\left({\frac{{{18}}}{{{38}}}}\right)}\cdot{\left({\frac{{{18}}}{{{38}}}}\right)}\right)}+{\left({\frac{{{18}}}{{{38}}}}\right)}\right]}$$
$$\displaystyle=-{3}{\left({.1458}\right)}+{0}+{\left(-{1}\right)}{\left({.2624}\right)}+{0}+{1}{\left({.5918}\right)}$$
$$\displaystyle=-{.108}$$
Now, going back to part b. This strategy is NOT a"winning strategy" because the expected winnings is a negative value, hence you really have an expected loss.

### Relevant Questions

The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with $$\displaystyle\mu={1.5}$$ and $$\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}$$.
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(e) What is the moment generating function for X?
The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius $$\displaystyle{R}={7.4}\times{10}^{{-{15}}}$$ m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
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When a gas is taken from a to c along the curved path in the figure (Figure 1) , the work done by the gas is W = -40 J and the heat added to the gas is Q = -140 J . Along path abc, the work done by the gas is W = -50 J . (That is, 50 J of work is done on the gas.)
I keep on missing Part D. The answer for part D is not -150,150,-155,108,105( was close but it said not quite check calculations)
Part A
What is Q for path abc?
Express your answer to two significant figures and include the appropriate units.
Part B
f Pc=1/2Pb, what is W for path cda?
Express your answer to two significant figures and include the appropriate units.
Part C
What is Q for path cda?
Express your answer to two significant figures and include the appropriate units.
Part D
What is Ua?Uc?
Express your answer to two significant figures and include the appropriate units.
Part E
If Ud?Uc=42J, what is Q for path da?
Express your answer to two significant figures and include the appropriate units.
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
The student engineer of a campus radio station wishes to verify the effectivencess of the lightning rod on the antenna mast. The unknown resistance $$\displaystyle{R}_{{x}}$$ is between points C and E. Point E is a "true ground", but is inaccessible for direct measurement because the stratum in which it is located is several meters below Earth's surface. Two identical rods are driven into the ground at A and B, introducing an unknown resistance $$\displaystyle{R}_{{y}}$$. The procedure for finding the unknown resistance $$\displaystyle{R}_{{x}}$$ is as follows. Measure resistance $$\displaystyle{R}_{{1}}$$ between points A and B. Then connect A and B with a heavy conducting wire and measure resistance $$\displaystyle{R}_{{2}}$$ between points A and C.Derive a formula for $$\displaystyle{R}_{{x}}$$ in terms of the observable resistances $$\displaystyle{R}_{{1}}$$ and $$\displaystyle{R}_{{2}}$$. A satisfactory ground resistance would be $$\displaystyle{R}_{{x}}{<}{2.0}$$ Ohms. Is the grounding of the station adequate if measurments give $$\displaystyle{R}_{{1}}={13}{O}{h}{m}{s}$$ and R_2=6.0 Ohms?
The following is an 8051 instruction: CJNE A, # 'Q' ,AHEAD
a) what is the opcode for this instruction?
b) how many bytes long is this instruction?
c) explain the purpose of each byte of this instruction.
d) how many machine cycles are required to execute this instruction?
e) If an 8051 is operating from a 10 MHz crystal, how longdoes this instruction take to execute?
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(a) What is the magnitude and direction of the force that theparaglider exerts on the earth ?
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The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
$$\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}$$
where A is the cross-sectional area of the vehicle and $$\displaystyle{C}_{{d}}$$ is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity $$\displaystyle\vec{{{v}}}$$. Find the power dissipated by form drag.
Express your answer in terms of $$\displaystyle{C}_{{d}},{A},$$ and speed v.
Part B:
A certain car has an engine that provides a maximum power $$\displaystyle{P}_{{0}}$$. Suppose that the maximum speed of thee car, $$\displaystyle{v}_{{0}}$$, is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power $$\displaystyle{P}_{{1}}$$ is 10 percent greater than the original power ($$\displaystyle{P}_{{1}}={110}\%{P}_{{0}}$$).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, $$\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}$$, is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
This problem is about the equation
dP/dt = kP-H , P(0) = Po,
where k > 0 and H > 0 are constants.
If H = 0, you have dP/dt = kP , which models expontialgrowth. Think of H as a harvesting term, tending to reducethe rate of growth; then there ought to be a value of H big enoughto prevent exponential growth.
Problem: find acondition on H, involving $$\displaystyle{P}_{{0}}$$ and k, that will prevent solutions from growing exponentially.