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 Numberon Board Probability 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 YardsGained 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 YardsGained ExpectedYardsGained 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?

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 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%.

### 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