MauiSunset’s Roulette Paradox

Copyrighted June 18, 2012, all rights reserved.

MauiSunset’s Roulette Paradox states: the chance of winning a “straight up” bet on an American Roulette wheel is NOT 1 in 38 but 1 in 52, resulting in a House Advantage 6 times as large, but with the payoff remaining the same 35 to 1.

Let me prove it to you.

The odds of winning a single number bet, a “straight up” bet, mathematically is defined as 1 divided by the number of slots on the Roulette wheel, 38 for an American wheel (numbers 1 – 36, 0 and 00) and 37 for a European wheel (numbers 1 – 36 and 0).  I don’t dispute this at all.  The odds are 1 in 38. (American wheel for the remainder of this post)

However, those odds are usurped by the probability of your number showing up in the next spin – a HUGE difference.  This is what the “Gamblers’ Fallacy” is all about – which demands that the odds MUST dominate the game of Roulette for EVERY spin.  If you see 10 Reds in a row and the odds are close to 50/50 then Black is grossly over due – that’s the Gamblers’ Fallacy, demanding the odds supersede probability.  The same applies here.

When the odds are calculated, 1 divided by 38, the assumption is that each and every one of those 38 slots is just as likely to be hit.  That’s wrong, that’s the Gambler’s Fallacy, demanding the odds dominate over statistics for EVERY spin.  In reality 1/3 of the numbers, actually 36.3%, on the Roulette wheel will not be included in the next spin and must be accounted for in the actual odds.

That would mean that if I pick a number, 0, 00, or 1 – 36, that it MUST show up in the next 38 spins – right?  Well that’s what it means when you say the odds are equal for every slot on the Roulette wheel for every spin.  If a duplicate number appears then the odds were not equal – that number was not 1/38 likely to hit – it just hit in the last spin.

In fact, if you bet on the same number for 38 times in a row, you will find that about 1/3 of these tests results with your number not being spun – that’s right about 1/3 of the time.

Roulette random numbers have a quirk – duplicates, triplicates, quadruplicates, and higher will spin out in the next 38 spins of the wheel.  I’ve included the mathematics proving this at the end of this article.  Basically when you have 38 numbers and 38 spins 1/3 (about 13-14 of them) will be spun once, 1/3 (13 -14 of them) will show up as duplicates and higher, and thus 1/3 (actually 36.3% or 13-14 of them) will not show up at all due to the dups.

Therefore the one number you pick up for “straight up” bets should show up in 38 + 14 or 52 spins, yet the payoff is still 35 to 1.

The House Advantage is calculated by the chance of winning which is 1/52 * 35 minus the chance of loosing which is 51/52 * -1 = -30.77% instead of the stated odds calculation of 1/38 * 35 – 37/38 * 1 = -5.26%.  The HA for Roulette Straight Up bets, in reality, is really 6 times more than theoretical odds calculation – it is a sucker bet.

As supporting proof I offer the fact that the average profit per Roulette table in Las Vegas NOT the House Advantage for Roulette of 5.26% but the tables produce a profit of 16% – 3 times as much profit, based on real spins and not theoretical odds.  Something must account for the HUGE profits of Nevada casinos and Straight Up bets is the answer; most of the bets on a Roulette table are Straight Up.  Nevada Roulette table winnings reported to the Nevada Gaming Commission:

In summary, betting on “straight up” single numbers is 6 times worse than the stated Roulette odds table; it’s a sucker bet.  The stated odds of 1 in 35 are wrong – the Gamblers’ Fallacy was used in that calculation, the probability of winning is really 1 in 52.

The “Observation of the Third of Random Numbers not spun”:

If the chance of spinning a single number is 1/38 then the chance of NOT spinning the number is 37/38 and for 38 spins is (37/38)^38 = 36.3%, that means for a Roulette wheel 38 * .363 = 13.8 or 14 numbers will not show up in the next 38 spins.

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