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Ask the Wizard #260

At a recent street fair, they had a game in which there was a field of numbers with shallow cups and a cup of balls and it involves addition. I didn’t ask the name of the game, and I searched the internet for about an hour but couldn’t find anything about it. I thought you might have some information about it, it’s odds, or at least the name.

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.

  1. 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.
  2. The player starts paying a specified fee per play, such as $1.
  3. The player will spill 8 marbles onto an 11 by 13 grid. Each marble will fall into one of the 143 holes.
  4. 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 Board
    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

  5. 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
    8 100
    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

  6. 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
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 × 10168, 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.


How many casinos are in the world?

Ben from Philippines

Using Casino City’s Pocket Gaming Directory as my source, I estimate there to be about 5,600.

I’d be interested in your thoughts about a poker dispute I observed at the Wynn. Here is the executive summary. One player called "all in" and pushed his stack of chips towards the pot. Another player called, and lost. The dealer started to count the chips, which included two $100 black chips hidden amongst a lot of $1 blue chips and $5 red chips. It turns out the first player had them at the bottom of his blue and red stacks. The second player argued he wouldn’t have called if he had known about the black chips. The Wynn ruled in favor of the first player, but the first player was furious about it. Did the Wynn make the right ruling?


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 2-5 NL and 5-10 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 no-limit 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 2-5 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.

At the Boulder Station, there is a side bet in craps called the "Replay" bet. It pays if the shooter makes the same point at least three times. If the shooter achieves a win on two or more different numbers, only the highest win is payable. I’m including the pay table. What are the odds on this bet?

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%.


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