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Ask the Wizard #260
Andrew from Queens, NY
The industry term for that game is Razzle Dazzle. I remember seeing it in southern California as a kid, and just last year in San Felipe, Mexico. It is usually skinned to look like a football game. This game is the worst of the carnival game scams, in my opinion. The state of New York should be ashamed for permitting it. Based on some research, the rules vary from place to place, but the gist of the con is always the same.
It is based on the same illusion as the field bet in craps. For those readers not familiar with the field bet, the player wins if the sum of the roll of two dice is 2, 3, 4, 9, 10, 11, or 12. Losing numbers are 5, 6, 7, and 8. Wins pay even money, except the 2 pays 2 to 1 and the 12 pays 3 to 1 (except at stingy Harrah’s casinos, where they pay 2 to 1 only on the 12). The mathematically challenged gambler may falsely reason it is a good bet because there are 7 totals that win and only 4 that lose. The reason the odds favor the house is that the losing numbers have the greatest chance to be rolled.
Here are the specific rules of Razzle Dazzle, as taken from the article Probabilities of Winning a Certain Carnival Game by Donald A. Berry and Ronald R. Regal, which appeared in the November 1978 issue of the The American Statistician.
 The object of the game is to advance across the football field 100 yards. The player will be awarded some kind of nice prize when he does.
 The player starts paying a specified fee per play, such as $1.
 The player will spill 8 marbles onto an 11 by 13 grid. Each marble will fall into one of the 143 holes.
 Each hole has a number of points from 1 to 6. The following table shows the frequency of each number of points.
Razzle Dazzle Points Distribution
Points Number
on BoardProbability 1 11 0.076923 2 19 0.132867 3 39 0.272727 4 44 0.307692 5 19 0.132867 6 11 0.076923 Total 143 1.000000  The total number of points will be added. The carnie will look up the point total on a conversion chart to see how many yards the player advances. The conversion chart is shown below.
Razzle DazzleConversion Chart
Points Yards
Gained8 100 9 100 10 50 11 30 12 50 13 50 14 20 15 15 16 10 17 5 18 to 38 0 39 5 40 5 41 15 42 20 43 50 44 50 45 30 46 50 47 100 48 100  If the player rolls a total of 29, then the fee for all subsequent rolls will be doubled, and the player be awarded one extra prize if and when he reaches the other end of the football field.
The average points per marble is 3.52, and the standard deviation is 1.31. Note how 3 and 4 points have the highest probability. That keeps the standard deviation low, and the sum of many marbles close to expectations. The standard deviation of the roll of a single die is 1.71, by comparison.
Next, notice how there are 20 winning totals and 21 losing totals on the yardage conversion chart. The kind of sucker who gambles on carnival games might incorrectly reason his probability of advance is 20/41 or 48.8%. It wouldn’t surprise me if the carnies falsely claimed these were the odds of advancing. However, much like the field bet, the most likely outcomes don’t win anything.
The next table show the probability of each number of points per turn, yards gained, and expected yards gained. The lower right cell shows the average yards gained per turn is 0.0196.
Expected Yards Gained per Turn
Points  Probability  Yards Gained 
Expected Yards Gained 
8  0.00000000005  100  0.00000000464 
9  0.00000000176  100  0.00000017647 
10  0.00000002586  50  0.00000129285 
11  0.00000022643  30  0.00000679305 
12  0.00000143397  50  0.00007169849 
13  0.00000713000  50  0.00035650022 
14  0.00002926510  20  0.00058530196 
15  0.00010234709  15  0.00153520642 
16  0.00031168305  10  0.00311683054 
17  0.00083981462  5  0.00419907311 
18  0.00202563214  0  0.00000000000 
19  0.00441368617  0  0.00000000000 
20  0.00874847408  0  0.00000000000 
21  0.01586193216  0  0.00000000000 
22  0.02642117465  0  0.00000000000 
23  0.04056887936  0  0.00000000000 
24  0.05757346716  0  0.00000000000 
25  0.07566411880  0  0.00000000000 
26  0.09221675088  0  0.00000000000 
27  0.10431970222  0  0.00000000000 
28  0.10958441738  0  0.00000000000 
29  0.10689316272  0  0.00000000000 
30  0.09677806051  0  0.00000000000 
31  0.08125426057  0  0.00000000000 
32  0.06317871335  0  0.00000000000 
33  0.04540984887  0  0.00000000000 
34  0.03009743061  0  0.00000000000 
35  0.01833921711  0  0.00000000000 
36  0.01023355162  0  0.00000000000 
37  0.00520465303  0  0.00000000000 
38  0.00239815734  0  0.00000000000 
39  0.00099365741  5  0.00496828705 
40  0.00036673565  5  0.00183367827 
41  0.00011909673  15  0.00178645089 
42  0.00003349036  20  0.00066980729 
43  0.00000797528  50  0.00039876403 
44  0.00000155945  50  0.00007797235 
45  0.00000023832  30  0.00000714969 
46  0.00000002632  50  0.00000131607 
47  0.00000000176  100  0.00000017647 
48  0.00000000005  100  0.00000000464 
Totals  1.00000000000  0  0.01961648451 
Here are some results of a random simulation of 17.5 million games.
Razzle Dazzle Simulation Results
Question  Answer 
Probability of advancement per turn  0.0028 
Expected yards gained per turn  0.0196 
Expected yards gained per advancement  6.9698 
Expected turns per game  5238.7950 
Average doubles per game  559.9874 
Averages prizes per game  560.9874 
I would have liked to indicate the average total bet per game, but my computer can not handle numbers so large. The average game had the player doubling his bet 560 times over the average of 5,239 turns per game. One game in the simulation had the player doubling his bet 1,800 times. Even at the average of 560 doubles, the bet per roll would be $3.77 × 10^{168}, assuming a starting bet of $1. That is many orders of magnitutude greater than the number of atoms in the known universe (source).
Even the most naive player will not play for long if he is advancing once every 355 plays only. What the carnies will do is cheat in the player’s favor at first. He may spot the player free rolls, or lie in adding up the points, giving the player winning totals to boost his confidence. I’ve never played the game, but I imagine that when the player gets close to the red zone (20 yards or less from a touchdown), then the carnie will start playing fairly. The player may wonder why he is suddenly getting nowhere, but with money already invested, and being so close to the goal line, he would hesitate to walk away and give up the yardage he already paid for.
Links
 Razzle Dazzle, excerpt from the book On the Midway.
 Razzle Dazzle Carny Board Game Arcade Scam.
 Probabilities of Winning a Certain Carnival Game by Donald A. Berry and Ronald R. Regal
Ben from Philippines
Using Casino City’s Pocket Gaming Directory as my source, I estimate there to be about 5,600.
ItsCalledSoccer
I’m not an expert on the fine points of poker rules, so I turned to David Matthews on this one. Here is what he says:
I play a lot of poker, mostly 25 NL and 510 NL. The rule is that oversized chips should be visible in front or on top, and if the dealer had known there were black chips, then the dealer should have requested that they be displayed. There is the conundrum of if the chips are hidden, how is the dealer going to see it? Stacks are also supposed to be one denomination. A stack of red ($5) chips on top of a $1 chip is considered a dirty stack because if someone were to eyeball the stack, they would likely guess the wrong number for the value. In this case, it would only be a $4 difference, but that’s the way it is. Interestingly, a stack of red chips with a $1 chip on top is not a dirty stack. I really think the dirty stack rules are a bit too much hassle.Whether the chips were visible or not is a real problem in nolimit hold ’em, because as is demonstrated in this situation, there can be confusion. Unfortunately, the ruling by the Wynn poker staff was the correct one, but it really was unfortunate for the person with the losing hand.
I have had a similar situation cost me $600. I went all in on a bluff against a guy, and he had a bunch of chips on top of some bills. $100 bills play in most places in town. I said to him, "What do you have? 2 Bills?" He just nodded and didn’t say anything. I went all in. He called instantly with 3 kings. He actually had 8 bills, and the house made me pay it. I would not have tried the bluff for an "All in" had I known how much money he had. That was expensive.
That’s why I ALWAYS ask on all ins. Even if a guy has 5 red chips ($25) and he throws it in, I ask the dealer how much it is. Dealers get irritated sometimes and look at me like, "It’s pretty obvious, isn’t it?" Also, players give me a hard time, too, sometimes. They tell me it’s obviously $100 or whatever it looks like. Doesn’t matter. I ask, "How much is it?" over and over again.
Another thing is I will usually bet a number rather than say "all in." If I had made a bet of $500 against the guy with the 3 kings, then it wouldn’t have mattered how many bills or what kinds of chips he had. I would have been on the hook for the $500 only.
I’m personally against having bills play on the table, because I constantly have to ask people how many bills. People get offended when you ask them over and over again, especially when they have less money on the table than other players, because they’re embarrassed to say, "2 bills." And then every hand I’m in with them I ask again, because you never know if maybe they added some bills to their stack in between hands. Or maybe they won a hand you didn’t see. In addition, just the fact that you ask someone how much they have might give away information about your hand.
I think paper should not play, and on 25 games and lower, the largest chips allowed in play should be $100 chips. My opinion isn’t a popular one, though.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
George from the Jungle
I checked, and indeed they do have that bet there. The return table below shows a house edge of 24.8%.
Replay
Event  Pays  Probability  Return 
4 or 10 four or more times  1000  0.000037  0.036892 
5 or 9 four or more times  500  0.000207  0.103497 
4 or 10 three times  120  0.000524  0.062847 
6 or 8 four or more times  100  0.000698  0.069815 
5 or 9 three times  95  0.001799  0.170927 
6 or 8 three times  70  0.004294  0.300609 
Loser  1  0.992441  0.992441 
Total  1.000000  0.247853 