Analyzing the dice game

Lifehacker had an interesting post today where they outlined a simple dice game where you have three distinct six-sided dice, each with a different number scheme. The neat thing was that the dice had a mutually intransitive set of win probabilities, similar to rock paper scissors. So if your opponent chooses first and you choose the die that has the 59 edge over theirs, you will win about 5% more of the rolls (118). What the author claims is that this results in a 98% chance of winning over the course of 20 rolls.

A commenter on the post though, commented that the average win percentage is only 61% as opposed to the 98% mentioned in the article. I was curious if this was true, so I whipped up a quick python script to test the theory:

import random

count = 1000000
rounds = 20

a_wins = 0
b_wins = 0
ties = 0

chance = 5.0 / 9.0

for i in range(count):
	a_score = 0
	b_score = 0

	for j in range(rounds):
		if random.random() < chance:
 			a_score += 1
 			b_score += 1

	if a_score > b_score:
		a_wins += 1
	elif a_score < b_score:
		b_wins += 1
		ties += 1

print '''
a wins %.2f%%
b wins %.2f%%
%.2f%% ties
''' % (100.0 * a_wins / count,
	100.0 * b_wins / count,
	100.0 * ties / count)

Essentially, run 1,000,000 trials, each with the specified number of rounds. Player A will score a point 59 of the time, so who will win?

Well here’s what running the script tells us:

a wins 60.92%
b wins 23.45%
15.62% ties

So it seems that thelongdivision’s post was completely correct. Player A (who knows the game and has the edge) has only a 61% chance of winning, rather than the 98% stated in the article–which also comes from the source article on Data Genetics. So what’s the difference. Are they just wrong or is there some error that both the other poster and I seem to have missed?

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