Demonstrate this.
But it is given in the OP as "p". The expectation takes the largest value for p = 1/2, because as post #6 shows the expected number of games played...
Let n = \frac{N-1}{2} \geq 0 Then our expectation value goes like this: E = \sum_{i=0}^{2i -1 < N} \sum_{j=0}^{2j -1 < N} { {i+j} \choose i } p^i...
Thus to decide a best-of-1 series, exactly one game is played taking us from state (0,0) to (1,0) with probability p or to state(0,1) with...
Expansion of (p+(1-p))^n \begin{array}{c} & & & & & & & 1 & & & & & & & \\ & & & & & & p & & (1-p) & & & & & & \\ & & & & & p^2 & & 2 p (1-p) & &...
Let N be an odd, positive number. A best-of-N series of games is played until one team has won a majority of the N maximum possible games in the...
This shows evidence of originating from a quote mine. First Edition was published circa 1998...
My flourless chocolate torte says you are wrong.
That "mechanism" question is literally not required to predict the behavior of gravitational phenomena, just as the color of your mother's KichenAid...
Not quite. The essential thing different with n>2 is that it is possible that the lead color changes. So as the number of colors goes down, so does...
Wrong. It's not a problem about how to paint the balls the same color. It's a question about how long a given description of the process is expected...
That's not actually the same problem. Here the balls are actually changing colors and nothing prevents any of the remaining colors from overtaking...
Excellent blog post by Laurent Lessard on this: http://www.laurentlessard.com/bookproofs/colorful-balls-puzzle/ A PDF by Tim Black:...
A Simpler Model nColors=15; transitions = DiagonalMatrix[1 - Table[Binomial[nColors + 1 - k,2]/Binomial[nColors,2],{k,1,nColors}]] +...
Automation All of the above can be coded and run for different numbers of balls from 1..15. Maybe higher if you don't use a free-to-use cloud...
Uh oh, where in simulation, each simulation individually came to an end, the state vector always has a non-zero component not yet in the final state....
Tricks with Matrices In addition to the five state transition matrix, lets have it automatically add 1 to a counter for the portion of our state...
Stochastic Matrixes and Simulation Instead of having sampling error, we could simulate the probability of outcomes. First we need to enumerate all...
Underthinking I predict a number of people will attempt simulation to attempt to model this. Simulation is reasonably fast to code, fast to run, and...
On April 28, FiveThirtyEight.com posted the weekend puzzle contest (which closed Sunday night) with their solutions coming this Friday, May the...
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