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Ask the Wizard #275

How do slot machine Community Bonuses work?

Anonymous

Community, or group, bonuses is a popular new concept in slot machines. While the bonus can be earned in different ways, depending on the game, the concept is that more than one player can play the same bonus round at the same time. Usually the outcome is shown on a large video screen visible to all players. One exception is the Wheel of Fortune community bonus, where an enormous wheel sits between the individual machines.

From what I can tell, the Wheel of Fortune community bonus is earned by an individual player achieving the initiating event on his own. It then starts a 10-second countdown, to give other players a chance to trigger a bonus. Usually this doesn't happen, so the player who earned the bonus plays by himself.

In contrast, on the American Idol slot machine, all active players play the community bonus. It is also triggered independently of where the reels stop, so no matter how many players are playing, or how fast they play, the odds of a bonus at any given time are always the same. The rest of this answer shall focus on how specifically the American Idol community bonus works.

It would be unfair if there were two players in a bank of community bonus machines, in which one was betting $1 a minute, and one was betting $10 a minute, if they both had the same expected value in the community bonus. To avoid such injustice, a multiplier is applied to the final bonus win, roughly according to the amount the individual player was betting at the time the bonus was initiated, factoring in both average bet and speed of play. How the game determines each player's multiplier is a bit complicated. Here is what I could figure out:

  • The game keeps track of the players' last 40 seconds of betting activity for each player, in the form of a queue.
  • Each 20 cents the player bets at one time buys him 10 seconds of 1x multipliers. So, a $1 bet would buy 10 seconds of 5x multipliers. This multiplier is added to the end of the player's queue.
  • If the player plays at a faster rate than one bet per ten seconds, causing there to not be enough space in the queue to add 10 seconds of multipliers, then it will cram any excess multipliers onto the last few seconds. How exactly it does this, I'm not sure.
  • As the player sits at the machine, the game will drop off from the stack the highest multipliers in the queue. Kind of like how in a queue for a Vegas nightclub it doesn't matter your position in line for girls, but how pretty you are; the most attractive girls are removed from the queue first.
  • As multipliers are dropped off the queue they are also the eligible multiplier should the community bonus be triggered at that moment.
  • At any given microsecond, except if a community bonus is already in progress, there is a fixed probability that the bonus will be triggered. So the time between bonuses would follow an exponential distribution. In other words, a bonus has a memoryless property and is never overdue, much like a royal flush in video poker.

Example: Let's say the player has 35 seconds of 1x multipliers in his queue. He then makes a $1 bet. Normally this would buy 10 seconds of a 5x multiplier. However, in this case there isn't enough room in his queue to add ten seconds. What the game would likely do is add four seconds of 1x multipliers, and one second of a 6x multiplier. I'm not saying it would do exactly this, but something along these lines, where the total time × average multiplier still equals 10.

Besides community bonuses, there are also "local bonuses" that are earned by each individual player and played only by that player. If a community bonus hits during a local bonus then that local bonus will be put on hold during the community bonus. To keep the local bonus player eligible for community bonus the game will give the player 60 seconds of multipliers. This is an exception to the usual 40-second queue. I am not sure how the multiplier during this time is determined. If the player finishes the local bonus in less than 60 seconds then the multipliers for the remaining time are somehow added to his queue.

One thing I have no idea about is what happens if a player sits down and starts playing during an ongoing community bonus, which the game will let him do. It would be unfair if the player were playing during this time without any chance to trigger another community bonus. I'm told that the machine somehow compensates the player in this situation, but I have no idea how.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

If you can shed any light on the unanswered questions in my answer, please feel free to write to me or post your comments in my forum.

The article Most-stolen cars: Old Hondas just released a list of the most stolen cars for 2010. What bothers me about this is that this information is of little use to the average person, who should be worrying about auto theft rates. Would you agree?

Anonymous

Absolutely, I agree! I think giving the public a list of the most stolen cars by the total stolen is at best giving the reader useless information, and at worst leading him to falsely conclude that an old Honda is more likely to be stolen than any other car.

What is useful information is to know is auto theft rates, in other words, cars stolen by the number on the road. An article that provides such rates is The 10 Most Stolen Cars in 2011. Most often stolen? The Cadillac Escalade. No Honda even makes their top 10.

Publishing a list of total stolen will unfairly worry mathematically challenged old Honda owners, when it is the Escalade owners who should be worrying. To every media source who quoted the list from the National Insurance Crime Bureau, and there were lots of them, I say shame on you!

What is your opinion of the lawsuit against the Quebec Lottery over allegedly non-random tickets?

Anonymous

For those not familiar with the story, the Quebec Lottery offers a game titled Extra. The machine randomly picks a 7-digit number, and the player has to match as many digits as possible, in order, from a random draw. The matching digits can be aligned in either direction. The smallest prize is $2 for matching the right-most digit only. The largest prize is $1,000,000 for matching all seven digits.

What the plaintiffs in the lawsuit noticed was that if you purchased ten Extras then for the first and last digits the game picked one of each numeral. In other words, if you looked at the first, or last, position only then you would see all ten numbers from 0 to 9. The plaintiffs claim this gives them only two chances to win and is non-random.

I see their point. Almost all the variance in that game comes from the $1,000,000 jackpot. The standard deviation of ten completely independent random tickets would be 1002.845. The way the Quebec Lottery does it the standard deviation of ten tickets purchased at the same time is almost the same at 1002.833.

In my opinion if the player buys multiple quick pick lottery tickets each ticket should be independent of the others. However, I find the lawsuit of $20 million almost totally frivolous. If I were the judge, I would award the plaintiffs $1.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

I was playing Hearts and was dealt 10 of them. What are the odds of that?

Anonymous

For those unfamiliar with the rules of Hearts, play starts with dealing 13 cards each to four players. The hearts suit is significant to the game, so how many you get is important. The following table shows the odds of being dealt 0 to 13 hearts.

Probability of 0 to 13 Hearts out of 13 Cards

Hearts Combinations Probability Inverse
13 1 0.0000000000016 1 in 635,013,559,600.0
12 507 0.0000000007984 1 in 1,252,492,228.0
11 57,798 0.0000000910185 1 in 10,986,773.9
10 2,613,754 0.0000041160601 1 in 242,950.8
9 58,809,465 0.0000926113531 1 in 10,797.8
8 740,999,259 0.0011669030492 1 in 857.0
7 5,598,661,068 0.0088166008164 1 in 113.4
6 26,393,687,892 0.0415639752774 1 in 24.1
5 79,181,063,676 0.1246919258321 1 in 8.0
4 151,519,319,380 0.2386080062219 1 in 4.2
3 181,823,183,256 0.2863296074662 1 in 3.5
2 130,732,371,432 0.2058733541286 1 in 4.9
1 50,840,366,668 0.0800618599389 1 in 12.5
0 8,122,425,444 0.0127909480376 1 in 78.2
Total 635,013,559,600 1.0000000000000

This question was raised and discussed in the forum of my companion site Wizard of Vegas.