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Slot Bingo
Introduction
Slot Bingo appears to be a pick-15 keno game. The game is made by SmartSoft Gaming, a provider of games for Internet casinos. As I show below, the game does not follow the natural odds of a fair keno.
After confronting SmartSoft with my analysis, they quickly removed the game.
Rules
- The player may play 1 to 8 cards at a time.
- Each card is marked with 15 random numbers, without replacement, from 1 to 80. The player may not choose his own numbers. The cards change after every game.
- After a bet is made, the game will draw 20 random balls, numbered 1 to 80, without replacement.
- If a ball drawn by the game matches a number on any card played, then it is marked on the card and it known as a "catch."
- Each card wins according to the number of catches and the pay table below.
Slot Bingo Pay Table
| Catch | Pays |
|---|---|
| 15 | 10000 |
| 14 | 1000 |
| 13 | 800 |
| 12 | 500 |
| 11 | 400 |
| 10 | 200 |
| 9 | 100 |
| 8 | 50 |
| 7 | 40 |
| 6 | 20 |
| 5 | 10 |
| 4 | 5 |
| 3 | 4 |
| 2 | 2 |
| 1 | 0 |
| 0 | 0 |
Analysis
The following table shows the expected return, assuming the balls are drawn fairly. Pays are on a "for one" basis.
Slot Bingo Return Table
| Catch | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| 15 | 10000 | 15,504 | 0.000000 | 0.000000 |
| 14 | 1000 | 2,325,600 | 0.000000 | 0.000000 |
| 13 | 800 | 137,210,400 | 0.000000 | 0.000017 |
| 12 | 500 | 4,310,693,400 | 0.000001 | 0.000325 |
| 11 | 400 | 81,903,174,600 | 0.000012 | 0.004937 |
| 10 | 200 | 1,009,047,111,072 | 0.000152 | 0.030412 |
| 9 | 100 | 8,408,725,925,600 | 0.001267 | 0.126716 |
| 8 | 50 | 48,650,485,712,400 | 0.007331 | 0.366572 |
| 7 | 40 | 198,344,287,904,400 | 0.029890 | 1.195589 |
| 6 | 20 | 572,994,609,501,600 | 0.086348 | 1.726962 |
| 5 | 10 | 1,168,909,003,383,260 | 0.176150 | 1.761501 |
| 4 | 5 | 1,660,382,107,078,500 | 0.250213 | 1.251066 |
| 3 | 4 | 1,595,269,083,271,500 | 0.240401 | 0.961604 |
| 2 | 2 | 981,704,051,244,000 | 0.147939 | 0.295878 |
| 1 | 0 | 346,917,972,996,000 | 0.052279 | 0.000000 |
| 0 | 0 | 53,194,089,192,720 | 0.008016 | 0.000000 |
| Total | 6,635,869,816,740,560 | 1.000000 | 7.721578 |
The lower right cell shows an expected return of 772%!
Experiment
A return of 772% seemed too good to be true, so I played the free game at the SmartSoft web site. I found the average number of catches to be ridiculously low. So I recorded 20 consecutive games for more careful analysis. The following video shows those 20 games.
The next table shows the number of catches on each card in each game.
Total Catches by Game and Card
| Game | Card 1 | Card 2 | Card 3 | Card 4 | Card 5 | Card 6 | Card 7 | Card 8 | Total |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 3 | 2 | 1 | 1 | 1 | 2 | 2 | 13 |
| 2 | 1 | 2 | 0 | 2 | 1 | 3 | 2 | 1 | 12 |
| 3 | 1 | 0 | 2 | 0 | 1 | 0 | 3 | 1 | 8 |
| 4 | 2 | 2 | 1 | 0 | 1 | 1 | 2 | 1 | 10 |
| 5 | 2 | 0 | 1 | 2 | 2 | 0 | 1 | 0 | 8 |
| 6 | 1 | 0 | 0 | 1 | 2 | 1 | 3 | 1 | 9 |
| 7 | 1 | 0 | 1 | 2 | 2 | 1 | 0 | 2 | 9 |
| 8 | 3 | 2 | 0 | 2 | 1 | 1 | 0 | 2 | 11 |
| 9 | 1 | 0 | 0 | 1 | 0 | 0 | 3 | 3 | 8 |
| 10 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 4 |
| 11 | 2 | 1 | 1 | 2 | 4 | 1 | 1 | 0 | 12 |
| 12 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 6 |
| 13 | 1 | 0 | 2 | 3 | 1 | 3 | 1 | 0 | 11 |
| 14 | 0 | 0 | 1 | 0 | 1 | 0 | 3 | 4 | 9 |
| 15 | 1 | 0 | 4 | 3 | 2 | 2 | 1 | 3 | 16 |
| 16 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 0 | 11 |
| 17 | 2 | 2 | 1 | 3 | 2 | 1 | 0 | 1 | 12 |
| 18 | 0 | 2 | 0 | 1 | 0 | 0 | 1 | 2 | 6 |
| 19 | 0 | 2 | 0 | 0 | 1 | 1 | 0 | 2 | 6 |
| 20 | 0 | 2 | 1 | 0 | 1 | 1 | 1 | 0 | 6 |
| Total | 23 | 20 | 19 | 24 | 28 | 20 | 27 | 26 | 187 |
The lower right cell shows a total of 187 catches.
In 20 fair games of 8 cards each with 15 numbers per card, the expected number of catches would be 20 × 8 × 15 × (20/80) = 600.
My results were 600-187 = 413 catches short of expectations.
The standard deviation of the total catches in 20 games is sqrt(20*8*15*(1/4)*(3/4)) = 10.606602.
This makes my results 413/10.606602 = 38.938013 standard deviations short of expectations.
In a fair game, the probability of 187 catches of less would be 1 in some number with about 330 digits. To put that in perspective, that would be like the chances of buying 39 random Powerball lottery tickets (each has a chance of winning of 1 in 292,201,338) and winning on all 39 of them.
You may be wondering if the game discloses that the ball draw is not fair. It does not that I can find. If you click the question mark in the game nothing happens.
It is easy to conclude either the cards or the ball draw is not random. I don't know exactly how they are doing it, but the numbers that appear least on the player cards are the most likely to be drawn.
Update
After giving SmartSoft Gaming an opportunity to respond to this page, they replied that Slot Bingo was a work in progress and not available for actual casino use. Furthermore, they indicated it was not a good idea to have incomplete games on their web site, which I see they removed.
I would like to commend SmartSoft for doing the right thing and removing the game.