Question-221052

Question Number 221052 by MrGaster last updated on 23/May/25

Answered by Frix last updated on 24/May/25

Black needs at least 2 rounds = 6 draws. He wins 16 out of 64 different games of 6 draws = 25% Red needs at least 2(2/3) rounds = 8 draws. He wins 24 out of 256 different games of 8 draws = 9.375% while black also wins plenty of these. Yellow needs at least 3(1/3) rounds = 10 draws. He wins 28 out of 1024 different games of 10 draws = 2.734% while both black and red also win plenty of these. ⇒ Although I didn′t calculate all possible games I still think black′s probability is the highest.

$$\mathrm{Black}\:\mathrm{needs}\:\mathrm{at}\:\mathrm{least}\:\mathrm{2}\:\mathrm{rounds}\:=\:\mathrm{6}\:\mathrm{draws}. \\ $$$$\mathrm{He}\:\mathrm{wins}\:\mathrm{16}\:\mathrm{out}\:\mathrm{of}\:\mathrm{64}\:\mathrm{different}\:\mathrm{games}\:\mathrm{of} \\ $$$$\mathrm{6}\:\mathrm{draws}\:=\:\mathrm{25\%} \\ $$$$\mathrm{Red}\:\mathrm{needs}\:\mathrm{at}\:\mathrm{least}\:\mathrm{2}\frac{\mathrm{2}}{\mathrm{3}}\:\mathrm{rounds}\:=\:\mathrm{8}\:\mathrm{draws}. \\ $$$$\mathrm{He}\:\mathrm{wins}\:\mathrm{24}\:\mathrm{out}\:\mathrm{of}\:\mathrm{256}\:\mathrm{different}\:\mathrm{games}\:\mathrm{of} \\ $$$$\mathrm{8}\:\mathrm{draws}\:=\:\mathrm{9}.\mathrm{375\%}\:\mathrm{while}\:\mathrm{black}\:\mathrm{also}\:\mathrm{wins} \\ $$$$\mathrm{plenty}\:\mathrm{of}\:\mathrm{these}. \\ $$$$\mathrm{Yellow}\:\mathrm{needs}\:\mathrm{at}\:\mathrm{least}\:\mathrm{3}\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{rounds}\:=\:\mathrm{10}\:\mathrm{draws}. \\ $$$$\mathrm{He}\:\mathrm{wins}\:\mathrm{28}\:\mathrm{out}\:\mathrm{of}\:\mathrm{1024}\:\mathrm{different}\:\mathrm{games}\:\mathrm{of} \\ $$$$\mathrm{10}\:\mathrm{draws}\:=\:\mathrm{2}.\mathrm{734\%}\:\mathrm{while}\:\mathrm{both}\:\mathrm{black}\:\mathrm{and} \\ $$$$\mathrm{red}\:\mathrm{also}\:\mathrm{win}\:\mathrm{plenty}\:\mathrm{of}\:\mathrm{these}. \\ $$$$ \\ $$$$\Rightarrow \\ $$$$\mathrm{Although}\:\mathrm{I}\:\mathrm{didn}'\mathrm{t}\:\mathrm{calculate}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{games} \\ $$$$\mathrm{I}\:\mathrm{still}\:\mathrm{think}\:\mathrm{black}'\mathrm{s}\:\mathrm{probability}\:\mathrm{is}\:\mathrm{the}\:\mathrm{highest}. \\ $$

Answered by vnm last updated on 25/May/25

I deleted the previous reply, the numbers were wrong there. These numbers were calculated in two different ways and the results are the same. The probabilities of winning for Yel Red and Blk are 0.183896106746, 0.311912000345, 0.504191892909.

$$ \\ $$$$\mathrm{I}\:\mathrm{deleted}\:\mathrm{the}\:\mathrm{previous}\:\mathrm{reply},\:\mathrm{the}\:\mathrm{numbers}\:\mathrm{were}\:\mathrm{wrong}\:\mathrm{there}. \\ $$$$\mathrm{These}\:\mathrm{numbers}\:\mathrm{were}\:\mathrm{calculated}\:\mathrm{in}\:\mathrm{two}\:\mathrm{different}\:\mathrm{ways}\:\mathrm{and}\:\mathrm{the}\:\mathrm{re}{s}\mathrm{ults}\:\mathrm{are}\:\mathrm{the}\:\mathrm{same}. \\ $$$$\mathrm{The}\:\mathrm{probabilities}\:\mathrm{of}\:\mathrm{winning}\:\mathrm{for}\:\mathrm{Yel}\: \\ $$$$\mathrm{Red}\:\mathrm{and}\:\mathrm{Blk}\:\mathrm{are} \\ $$$$\mathrm{0}.\mathrm{183896106746}, \\ $$$$\mathrm{0}.\mathrm{311912000345}, \\ $$$$\mathrm{0}.\mathrm{504191892909}. \\ $$

Commented by vnm last updated on 25/May/25

((333778838716398691656)/(1815040267152105467903)) ((5095195563912530246309)/(16335362404368949211127)) ((8236157292008830739914)/(16335362404368949211127))

$$ \\ $$$$\frac{\mathrm{333778838716398691656}}{\mathrm{1815040267152105467903}} \\ $$$$\frac{\mathrm{5095195563912530246309}}{\mathrm{16335362404368949211127}} \\ $$$$\frac{\mathrm{8236157292008830739914}}{\mathrm{16335362404368949211127}} \\ $$

Commented by Frix last updated on 25/May/25

We′re interested in the path. How did you calculate these numbers?

$$\mathrm{We}'\mathrm{re}\:\mathrm{interested}\:\mathrm{in}\:\mathrm{the}\:\mathrm{path}.\:\mathrm{How}\:\mathrm{did}\:\mathrm{you} \\ $$$$\mathrm{calculate}\:\mathrm{these}\:\mathrm{numbers}? \\ $$

Commented by vnm last updated on 25/May/25

https://pastebin.com/n32saeVn Here′s a programme that calculates these numbers. To test it you can write a simple programme that finds the same numbers statistically using a random number generator. To get these numbers as fractions we need to create a class to work with long integer numbers.

$$ \\ $$$$\mathrm{https}://\mathrm{pastebin}.\mathrm{com}/\mathrm{n32saeVn} \\ $$$$\mathrm{Here}'\mathrm{s}\:\mathrm{a}\:\mathrm{programme}\:\mathrm{that} \\ $$$$\mathrm{calculates}\:\mathrm{these}\:\mathrm{numbers}.\:\mathrm{To} \\ $$$$\mathrm{test}\:\mathrm{it}\:\mathrm{you}\:\mathrm{can}\:\mathrm{write}\:\mathrm{a}\:\mathrm{simple}\:\mathrm{programme}\: \\ $$$$\mathrm{that}\:\mathrm{finds}\:\mathrm{the}\:\mathrm{same}\:\mathrm{numbers}\:\mathrm{statistically}\:\mathrm{using}\:\mathrm{a}\:\mathrm{random}\:\mathrm{number}\:\mathrm{generator}.\: \\ $$$$\mathrm{To}\:\mathrm{get}\:\mathrm{these}\:\mathrm{numbers}\:\mathrm{as}\:\mathrm{fractions}\:\mathrm{we}\:\mathrm{need}\:\mathrm{to}\:\mathrm{create}\:\mathrm{a}\:\mathrm{class}\:\mathrm{to}\: \\ $$$$\mathrm{work}\:\mathrm{with}\:\mathrm{long}\:\mathrm{integer}\:\mathrm{numbers}. \\ $$

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