The Japanese game of go is played with white and black pieces, called stones, on a board similar to checkers, with a grid of 19 by 19 intersections. The goal of this game is not to capture an opponent’s pieces, but to secure territory. The rules are simple, but the game is very challenging and many people think that it is much more difficult than chess.

A story problem

A Japanese-American multi-billionaire who was an aficionado of go and who wanted to make it more popular, designed the following lottery, open to anybody in the whole world.

In order to enter you only have to go to a specially designed secured web
site and fill a 19 by 19 go board with white and black dots representing
stones. (On each intersection you make either a white or a black dot.) The
entry is completely free, and filling out an entry takes approximately 10
minutes. You may fill out as many boards as you like. At s specified date
a configuration of white and black stones will be created randomly on a
board in a way similar to the method used in Powerball. *Anyone who
has correctly predicted this random configuration wins ten billion US
dollars.*

The news about the lottery spread all over the world and approximately 750 million entries were registered.

Answer the following questions:

1. Do you think that there were any winners? If so, how many?

2. What is the expected number of winners as a function of the number of
different entries?

3. Would you enter this lottery? Why? Why not? If you entered, how many
boards would you fill out? What would be your chance (probability) of
winning?

Be prepared to justify your answers.

Answer to 2. What is the expected number of winners when there are 750,000,000 entries? when there are E entries?

The probability that one entry will win is

1/(2^{(19*19)}= 1/(2^{361}).

First we want to know the probability that there will be a winner when
there are 750*10^{6} entries. That number is

(750*10^{6})/ (2^{361}) = A/B.

Let’s simplify it. We use a TI-83/84 calculator to help with the
calculations.

2 =10^{log(2)}

2^{361}= (10^{log(2)})^{361 }^{}10^{log(2)*361}
= B (the denominator of our expression above)

750*10^{6} = 10^{?} =10^{log(750)*106} =10^{log(750)+6}
= A (the numerator of our expression above)

A/B = 10^{log(750)+6}/10^{log(2)*361}

= 10^{-log(2)*361+ log(750)+6}

The exponent -log(2)*361+ log(750)+6 approximately equals -99.79676717 »
-100

So A/B » 10-100 =1/10100.

When there are E entries rather than 750,000,000, we substitute log(W) in
expression A above for log(750)+6.

So A/B becomes

10^{log(W)}/10^{log(2)*361} = 10^{log(W)}/10^{log(2)*361}
= 10^{log(W)}/10^{log(2)*361} = 10^{log(W)}/10^{log(2)*361}

10^{log(W)}/10108.6718.

The population of our planet is approximately 7.242 billion people. If everyone entered the contest above and filled out a board with black and white dots, what is the probability that there would be a winner?

log(7.242*10^{9}) = 9.86.

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