I've been preoccupied with the thread on math skills and multiple Place Bet calculations. I’ve done a little digging and have come up with some interesting bits that I’d like to share at the risk of being assigned a seat on the short bus. To avoid that I’ll try my best to offer somewhat sound reasoning and support from sources far more skilled in math than me. I cannot claim I’m correct but I have found enough to make me wonder. This is really a mental exercise for anyone interested.
The way we calculate HA on Place Bets. The 6 or 8 can win 5 ways and lose 6 ways making 11 events that can affect it. If we bet either the 6 or 8 for $6 there is $66 invested in the number of possible events. 5 winning events = $65 on the 5 wins (5x (the $7 won & $6 bet returned)). $66-$65 = $1 & 1/66 = 1.515%.
To start and recap I’ll repeat what I said regarding Zeke Feinberg’s multiple Place Bets calculations. Zeke said that when Place Bets were combined the House Advantage lessened based on:
Bet 6 & 8. 10 ways to win and 6 ways to lose making 16 events that can affect it. Investment is $12 ($6 + $6) x 16 = 192. 10 winning events = $190 = (($7 won + $12 bet returned) x 10). $192 - $190 = $2 & 2/192 = 1.042%
Bet Inside. 18 ways to win and 6 ways to lose making 24 events that can affect it. Investment is $528 ($5 + $6 + $6 + $5) x 24. 18 winning events = $522 = ($7 won + $22 bet returned) x 18). $528 - $522 = $6 & 6/528 = 1.136%
Bet Across. 24 ways to win and 6 ways to lose making 30 events that can affect it. Investment is $960 ($5 + $5 + $6 + $6 + $5 + $5) x 30. 24 winning events = $948 = $9 won 6 times ($54) + $7 won 18 times ($126) + ($32 bet returned) x 24) ($768). $960 - $948 = $12 & 12/960 = 1.25%
My response was: “The problem with Zeke's seductive math is that the numbers are being combined as a single bet when in fact they are independent. One number can win without any affect on the other even though both can lose. Each number needs to be calculated for an independent win even though a loss wipes out all bets. Investment on each number is $6 on each of 5 ways to win ($30) plus 6 ways to lose ($36) or $66 on 6 and $66 on 8 for a total of $132.
The number of events really should be 11 and not 16 since only one of the two numbers can actually win while the other is unaffected ($12 x 11 = $132). But I must say Zeke's math does look so logical.”
In spite of my answer this continued to gnaw at me. I found a 7/25/15 article by Jerry Stickman at
http://stickman.casinocitytimes.com/art ... side-64374
that pretty much backs up my response. I also found another Stickmen article from 5/18/07 at
http://stickman.casinocitytimes.com/art ... raps-34389
that provides a nifty ways to calculate combined bets while maintaining their independence. Using that method Inside is 2.597% and Across is 3.75%.
Just so we are all looking at this the same way the combined bet must be treated as a single bet. You cannot add to or regress any individual number unless you do so equally to all those numbers. If the bet is removed you do so to all the numbers in the combined bet. Under these conditions the player certainly has acquired more ways to win than lose.
I understand that doesn’t cut it for many math people but then I came across 12/6/10 article by Al Krigman at
http://krigman.casinocitytimes.com/arti ... raps-59413
That pretty much lines up with Zeke’s take. So even the experts can’t agree?
Then I came across the Place Bet section in Scarne’s New Complete Guide To Gambling, page 332 – 333 that addresses this very question. In addition there is the 10/2/2000 Al Krigman article at
http://krigman.casinocitytimes.com/arti ... guise-5541
that arrived at the same conclusion as Scarne. Combined Place Bets are actually Lay Bets in disguise. Someone who has only had vanilla ice cream their entire life might believe ice cream can only be vanilla. Many are going to say that a Lay is betting that a 7 will show before a specific chosen number. That’s what the game tells us but perhaps the math doesn’t care about the game, just the math. In the simplest terms a Lay is betting with the odds in your favor that another event will not occur. This friendly situation means betting more than you’ll win and the accepted casino version of a Lay means tacking on vig for casino profit.
If you work out the combined Place bets as Lay Bets the percentages line up as Zeke calculated them but for a different reason. They provide more ways to win than lose; you risk more to win less but the 7 is still not your friend.
Regular Lays as calculated at Wizard of Odds site.
Lay bet to lose on 6 or 8: [(6/11)×19 + (5/11)×(-25)]/25 = (-11/11)/25 = -1/25 = -4.000%
Lay bet to lose on 5 or 9: [(6/10)×19 + (4/10)×(-31)]/31 = (-10/10)/31 = -1/31 = -3.226%
Lay bet to lose on 4 or 10: [(6/9)×19 + (3/9)×(-41)]/41 = (-9/9)/41 = -1/41 = -2.439%
Combined Place Bets Calculated As Lays
Lay bet on 6 & 8: {(10/16)X7+(6/16)X-12]/12 = -.125/12 = -1.04%
Lay bet on 5 & 9: {(8/14)X7+(6/14)X-10]/10 = -.2857/10 = -2.857%
Lay bet on 4 & 10: {(6/12)X9+(6/12)X-10]/10 = -.5/10 = -5%
Inside & Across Combined Place Bets Calculated As Lays
Lay bet on Inside: {(18/24)x7+(6/24)X-22]/22 = -.25/22 = -1.136%
Lay bet on Across: {(6/30)x9+(8/30)x7+(10/30)x7+(6/30)X-32]/32 = .401/32 = 1.25%
As far as combining the Lay bets I point to the 7/28/97 Al Krigman article at
http://krigman.casinocitytimes.com/arti ... craps-5367
Krigman says that combining casino accepted Lay Bets moves the advantage back to the casino as the player accepts more ways to lose. I find it hard to believe that combining casino Lay Bets benefits the casino while no such benefit is derived by the player for combining Place Bets. That just doesn't seem logical if the conditions I laid out hold true.
I found this to be interesting but maybe that’s just me.
Kelph