Dhamnekar Win,odd

Get your friends and family to play this simple little game that illustrates a key aspect of stochastic mathematics in application to finance. 1. Draw on a big sheet of paper a sequence of 30 squares and label them consecutively 0 (bankrupt), 1, 1, 2, 2, 3, 3, 4, . . . , 14, 14, 15 (millionairedom). Each of these squares represents one state in a 30-state Markov chain. Imagine each state represents the value of some asset such as the value of a small business that each player is managing. 2. Give each player a token and a six-sided die. 3. At the start place each token on the second “2”, the fifth state. Imagine this corresponds to the small business having an initial value of $200,000. 4. Each turn in the game corresponds to, say, one year in time. In each year the business may be poor or may grow. Thus in each turn each player rolls his/her die and moves according to the following rules: • If a player rolls a 1 or 2, then he/she moves down some states; • if a player rolls a 3, he/she stays in the same state; • if a player rolls a 4, 5, or 6, he/she moves up some states. But the number of states (squares) a player moves is given by the number written in each square. Thus in the first move, because the fifth square/state is a “2,” a player moving up moves from the fifth square to the seventh square, and a player moving down moves to the third square. That the number written in each square is (roughly) proportional to the position of the square in the sequence corresponds to the financial reality that small businesses usually grow/shrink by small amounts, whereas large companies grow/shrink by large amounts. Investors expect returns in proportion to their investments. 5. Each player continues to role his/her die and move until reaching 0 or 15. That is, players continue to operate their businesses until they either go bankrupt or reach millionairedom. Questions: 1. Why do you expect each business to grow? That is, why do you expect each player to reach the “millionairedom” state? 2. When you play the game, roughly what proportion of players reach millionairedom? What proportion go bankrupt? 3. How do you explain the actual results? How would you answer all the above three questions? What are your answers to all of these above three questions? How can I write a program in R or in Octave to simulate this game?

In a chess tournament, each participant plays with every other participant exactly once. Each participant gets 1 point for a winning a game, 0.5 points for a draw and no point for a loosing a game. At most, how many of the 40 participants can score 24 points or more? My attempt to answer this question: There will be [math]\binom{40}{2}=780[/math] games in this chess tournament. Now each player will play 39 game. The probability of winning a game is [math]\frac13[/math]. Probability of loosing a game is [math]\frac13[/math] and the probability of making a game draw is [math]\frac13[/math]. So, out of 780 games, 260 games will be won by some players, 260 games will be lost and 260 games will be drawn. How can we proceed from here to answer this question?

[math]\displaystyle\int\limits_C xydx + (z-x)dy + 2yzdz = 0-2+8=6[/math]

Calculate the line integral of the vecor field f(x,y,z) along the curve C [math]\left(\displaystyle\int\limits_C f \cdot dr\right) [/math] 1)f(x,y,z) = yzi + xzj + xyk; C : the polygonal path from (0,0,0) to (1,0,0) to ( 1,2,0). 2)f(x,y,z) = xyi + (z-x)j + 2yzk; C : the polygonal path from (0,0,0) to (1,0,0) to (1,2,0) to (1,2,-2). How can I use here Stokes theorem? My work to answer to both these questions is in progress. Any math help, clue, or even correct answers will be accepted.😇😀

A circular target of unit radius is divided into four annular zones with outer radii 1/4, 1 /2, 3/4, and I, respectively. Suppose 10 shots are fired independently and at random into the target. (a) Compute the probability that at most three shots land in the zone bounded by the circles of radius 1 /2 and radius 1. (b) If 5 shots land inside the disk of radius 1 /2, find the probability that at least one is in the disk of radius 1 /4. My answers:(a) [math] \displaystyle\sum_{k=0}^{3}\binom{10}{k}(\frac34)^k (\frac14)^{10-k}[/math] (b)[math] \frac{\displaystyle\sum_{k=1}^{5}\binom{5}{k}(\frac{1}{16})^k(\frac{3}{16})^{5-k}}{\binom{10}{5}(\frac14)^5(\frac34)^5}=1.275e-2[/math] My answer to (a) is correct. Author's answer for (b) is [math] 1- (\frac34)^5[/math] Whose answer is correct? My answer for (b) or author's answer for (b)?

It is sometimes necessary to compute E° for a given half reaction from other half reaction of known E°. For example, the standard electrode potential for the oxidation of Titanium metal to Ti³⁺ can be obtained from the following half reaction: Ti²⁺ + 2 e⁻ ⇌ Ti⁰ E° = -1.63 Volts ① Ti³⁺ + e⁻ ⇌ Ti²⁺ E° = -0.37 Volts ② Calculate E° for Ti⁰ ⇌ Ti³⁺ + 3 e⁻ My answer: ② is more spontaneous reaction than ①. Reduction: Ti³⁺ + e⁻ ⇌ Ti²⁺ E° = -0.37 Volts ; Oxidation: Ti²⁺ + 2 e⁻ ⇌ Ti⁰ E° = -1.63 Volts Hence, E∘rx=E∘red−E∘ox=−0.37V−(−1.63V)=+1.26VErx∘=Ered∘−Eox∘=−0.37V−(−1.63V)=+1.26V My answer matches with author's answer. But, Is my logic in answering this question correct?

Is the above intuitive explanation of "why the inequality [math] P(A_2|A_1) > P(A_1)[/math] is true" correct?

My answer to (a) N= Total symbols on each page. r= 50 misprints. So, what is impossible is two or more misprints occurring in the same symbol. So, first condition of Fermi-Dirac statistics is satisfied. All distinguishable arrangements satisfying the first condition have equal probabilities. In our case, It is [math]\binom{500N}{50}^{-1}[/math]

Suppose that coin 1 has probability 0.7 of coming up heads, and coin 2 has probability 0.6 of coming up heads. If the coin flipped today comes up heads, then we select coin 1 to flip tomorrow, and if it comes up tails, then we select coin 2 to flip tomorrow. If the coin initially flipped is equally likely to be coin 1 or coin 2, then a)what is the probability that the coin flipped on the third day after the initial flip is coin 1? b)Suppose that the coin flipped on Monday comes up heads. What is the probability that the coin flipped on Friday of the same week also comes up heads? My attempt to answer this question:

If f is in the direction of r'(t) then it is in the direction of increasing r(t). So, the directions of f(t) and r'(t) will also be same. Definition of dot product: And ds =|| r'(t)|| ⇒ [math]\sqrt{x'(t)^2 + y'(t)^2}dt [/math] so, work = force × distance = [math] \displaystyle\int_C f\cdot dr= \displaystyle\int_C \left\vert\vert{f} \right\vert\vert ds[/math]

. Show that if f points in the same direction as r'(t) at each point r(t) along a smooth curve C, then [math]\displaystyle\int_C f\cdot dr = \displaystyle\int_C ||f|| ds ?[/math] How can I prove this corollary in line integral?

I got the answer to this question. I computed it as follows: [math]\displaystyle\int_0^{2\pi} 12 dt = 24\pi[/math]

Use a line integral to find the lateral surface area of the part of the cylinder [math]x^2 +y^2 =4[/math] below the plane x + 2y + z =6 and above the xy-plane. How to answer this question? What is the answer of this question? Graph of the required lateral surface area of the cylinder is given below:

I tag this question as 'SOLVED'. Author's computations are correct. Thank you.

After studying these author's computations, I understood the differentiation part but I don't understand cancellation part and final term computation.

Author computed [math]F_{X_k} (x)[/math] as follows. but I don't understand it. Would any member explain me the following computations?

Answers to all of the aforesaid questions are affirmative.

I think my answer in the original post is incorrect. So, I want to correct it. My corrected answer is [math] \displaystyle\int_0^{2\pi}\displaystyle\int_0^2\displaystyle\int_0^4 r dz dr d\theta =16\pi [/math] Is this answer correct?

Find the volume of the solid bounded by [math]z = x^2 + y^2 [/math] and [math] z^2 = 4(x^2 + y^2) [/math] My attempt to answer this question: [math] \displaystyle\int_0^{2\pi}\displaystyle\int_0^2\displaystyle\int_0^4 1 \rho^2 \sin{(\phi)} d\rho d\phi d\theta = -\frac{128\pi \cos{(2)}- 128\pi}{3}= 189.822143919[/math] Is this answer correct?

I computed the volume inside both the sphere [math] x^2 + y^2 + z^2 =1[/math] and cone [math] z= \sqrt{x^2 + y^2}[/math] as follows: [math]\displaystyle\int_0^{2\pi} \displaystyle\int_0^{\frac{\pi}{4}}\displaystyle\int_0^1 \rho^2 \sin{\phi}d\rho d\phi d\theta= 0.61343412301 =\frac{(2-\sqrt{2})\pi}{3}[/math]. This answer is correct.

I want to correct the volume asked in the question computed by me = [math] \frac{\pi}{3} \times \frac12 \times \frac{1}{\sqrt{2}} + \frac{\pi}{3} \times (1-\frac{1}{\sqrt{2}})^2 \times (\frac{3}{\sqrt{2}} -(1-\frac{1}{\sqrt{2}}))=0.534497630798[/math]

My attempt: I graphed the cone inside the sphere as in my first post. But I don't understand how to use the change of variables technique here to find the required volume. My answer without using integrals is volume of the cone + volume of the spherical cap = Is this answer correct? If correct, how to derive this answer using integration technique?

Find the volume V inside both the sphere x2 + y2 + z2 = 1 and the cone z = [math]\sqrt{x^2+ y^2}[/math] How to use change of variables technique in this problem?