variance for (six different) even money bets in French Roulette

Everybody knows that bets in roulette have different variance depending on the payout. However, after diving deeper I discovered that there are six essentially different even money bets in French Roulette, which has a house edge of 1/74 or 1.35%. There are two reasons for that. First, roulette layout has imperfect distribution of possible outcomes if you split your bet into two or three parts. Second, the very fact that you can bet everything on even money (any of 6 bets, namely Low, High, Odd, Even, Red or Black) but also on any combination of 2 or 3 "sub-bets". To make this easier to understand:
type 1 "Double or Nothing" - any bet from 6 possible (L/H/E/O/R/B)
type 2 "Double Fun" - any combination of 2 even money bets (12 possible - Low-Even, Low-Black, Low-Odd, Low-Red and so on)
type 3 "Triple Fun" - any combination of 3 even money bets (8 possible - LEB, LER, LOB, LOR, HEB, HER, HOB, HOR)
Of course, most popular "Double or Nothing" bet is mathematically the same in each of its six incarnations (18 chances to lose, 18 chances to double, 1 chance to win half). However, for Double Fun there are three different bets and two for Triple Fun. Long story short, the complete table look like this:

https://drive.google.com/file/d/129P7nbArPOrQsRPsXBtQ3UOQxHLEEw33/view?usp=sharing

I have a question for the community what is the variance for all these six bets and one for the Wizard - what is the highest variance possible? Bear in mind, even with £/$60 many combinations are possible, for example 45 on Red, 10 on Low and 5 on Even. The number of combinations is theoretically infinite and proof which pattern has highest (and lowest) possible variance may need a clever approach.

Here are my variances, based on a total bet of 1 unit:

Return >>>	0	1/2	2/3	1	4/3	2	Total	Variance
Even money only any of 6	18	1	0	0	0	18	37	0.980
2 x 30 any High or Low	9	1	0	18	0	9	37	0.493
2 x 30 ER or OB	8	1	0	20	0	8	37	0.439
2 x 30 EB or OR	10	1	0	16	0	10	37	0.547
3 x 20 in LER or HOB (left or right side)	4	1	14	0	14	4	37	0.307
3 x 20 in LOR or HEB (every second)	5	1	13	0	13	5	37	0.355

Thank you very much! I may have discovered casino games with record low s.d., below Pai Gow and Pai Gow Poker! :-) This may mean that even money bet (any of 6) has the highest variance possible hence Double Fun and Triple Fun (and any other combination) can only decrease it. I would love if Wizard could confirm that.

Quote: MattUK

Thank you very much! I may have discovered casino games with record low s.d., below Pai Gow and Pai Gow Poker! :-) This may mean that even money bet (any of 6) has the highest variance possible hence Double Fun and Triple Fun (and any other combination) can only decrease it. I would love if Wizard could confirm that.

I’m not the Wizard, but here is an element of answer.

What you call Double -or Triple- Fun is analog to playing two -or three- simple bets in consecutive rounds, except there is a correlation. But that correlation is very small, it is ‘almost’ independent. So looking at independent rounds gives a useful insight.
Elementary prob theory teaches that two (resp. three) independent bets at 30 have half (resp. one third) the variance than one bet at 60 .

—

Please note the slight difference in ChesterDog’s table for Hi or Lo, where both bets are independent. This is due to the treatment of the Zero event: he assumes you get back half your wager, while in some casinos your bet is ‘’in prison’’, i.e. you repeat your bet. In both cases, it is not the exact same bet, so variance is a bit over half the ‘simple bet’ variance (0.9795).

If you put all your bankroll at once on a single number, but for an EV of -2.7% :

Return >>>	-1	+35	Total	Variance
All on one number	36	1	37	0.9211

Why is it lower? Because the EV is farther from zero. In American roulette, it would be the highest variance.

Conclusion —> It is not variance or EV or House edge that one must look at, it is the RATIO OF STANDARD DEV ON EXPECTATION.
American : play all on one number. French : play all on an even bet.

I don't know where you've got your variance from. For even money bet on European Roulette it's 4*[(18/37) - (18/37)^2] = 0.9993. The rest in equally unhelpful. The question(s) still stand - what is the highest and lowest possible variance in French Roulette (1.35% house edge, 6 basic types and almost infinite combinations between them). So far we know that French Roulette can have s.d. both higher and lower than Pai Gow Poker, which is pretty amazing.

Quote: MattUK

I don't know where you've got your variance from. For even money bet on European Roulette it's 4*[(18/37) - (18/37)^2] = 0.9993. The rest in equally unhelpful. The question(s) still stand - what is the highest and lowest possible variance in French Roulette (1.35% house edge, 6 basic types and almost infinite combinations between them). So far we know that French Roulette can have s.d. both higher and lower than Pai Gow Poker, which is pretty amazing.

Look, it may not help YOU, but that does not mean it is unhelpful.
I explained the rain to a simple girl. She kept staring at my finger instead of the clouds it was pointing to, and she said ‘’it is unhelpful’’. So I made a drawing.

The question HAS been answered. In French roulette, limited to your six types, the highest variance is ‘all on one type’ ; the lowest is ‘divide in three on HOB or LER’.

Variance is E[X^2] - E[X]^2 =

• 1 - (1/37)^2 = .9993 for Eur even money
• (36.25/37) - (1/74)^2 = .9795 for Fr even money
• (1225+36)/37 - (1/37)^2 = 34.0804 for Eur=Fr single number bet
• For the triple fun HOB, consider betting h on High/Passe, b on Black and (1-h-b)=o on Odd, for a total wager of 1 unit.
  
  

  Return >>>	0	1/2	2h	2b	2o	2(h+b)	2(h+o)	2(b+o)	2	Total
  Proba (x1/37)	4	1	4	5	5	5	5	4	4	37

  
  Variance = 72.25/37-(72/37)h(1-h)-(80/37)bo - (73/74)^2

Minimizing this gives h=4/13, b=o=4.5/13 for a variance of 0.30595


Hope this helps...

Ach, now I'm with you. Yes, ChesterDog kindly calculated variance for all six types. We now know which of them has the highest and lowest. But that only leads to question what is the lowest and highest possible (very likely these). One way to solve this would be to brute force each and every combination with £/$ 60 bet, 5 minimum (all 6 "Double or Nothing" cases etc). Then arrange the variance from highest to lowest and the answer would reveal itself (through it's not a proof in the strict sense, but good enough).

On a side note, I think it would be marvelous if Wizard team could add the variance to any bet in demo mode. AFAIK no casino software provider currently offers that, even as an option. So far the house edge comparison page has unassuming note e to roulette: "Standard deviation depends on bet made". We now know that it's anything from 0.55 to 5.84.