Probability Of A Slot Machine

First of all, we must start with the number of possible combinations. In the case of slots, it is relatively simple – just multiply the numbers of symbols on each reel. The oldest slots had, for example, 3 reels with ten different symbols on each. The total number of combinations that could appear on the panel was 1,000 (10 x 10 x 10).

The number of combinations in today’s slots is somewhat higher. If we assume five reels with 30 symbols on each, we get a total of 243,000,000 combinations.

If you want to calculate your chances to win on an online slot machine, all you need is this simple equation:

Number of winning combinations / Total number of combinations

A person is doing 3 trials in 2 slot machines A and B. The chance of winning in machine A is 0.4 and in B is 0.2. In the first trial, this person choose the machine at random. In each stage, if he wins, he would again choose the same machine, else he chooses the other machine. 1) what is the probability that the 2nd trial is machine A? The probability of you getting the jackpot is exactly the same, however, whoever is watching that machine could easily make it so that you only got it one in every one-hundred-thousand times. This computer chip method is used in modern slot machines today.

A slot machine house edge is known by casino managers as the “hold”, and hold percentages vary a great deal, and do tend to be smaller at more expensive slots, frequently found to be around 1- 3% at the five dollar slots. Dan from Communication Group 34 takes you through the rules and mathematics of Casino slot Machines!Hub Video:

To calculate the payout of the slot machine, modify the formula a little:

Σ (winning combination_k * possible yield_k) / (Total number of combinations)

Let’s analyze a few basic slot machines. For the purposes of our article and in order to simplify the calculation, we will assume that the slot machine has only one payout line and the bet is one coin per round.

Analysis of the simplest slot machine

Let’s go back to the past and assume that the machine only has 3 reels and there is an apple, an orange, a lemon, a banana, a melon and a joker symbol on each. The individual combinations produce these winnings:

  1. Three jokers win 30 coins
  2. Any three fruits win 10 coins
  3. Two jokers win 4 coins
  4. One joker wins 1 coin

The total number of combinations is 216 (6 x 6 x 6).

Total number of winning combinations:

  1. In the first case there is only one winning combination (1 x 1 x 1 = 1)
  2. In the second case we have 5 winning combinations (3 times apple or 3 times orange or 3 times lemon, …) (1 x 1 x 1) x 5 = 5
  3. The joker may appear on any two reels. The calculation is as follows: 1 x 1 x 5 + 1 x 5 x 1 + 5 x 1 x 1 = 15
  4. The joker may appear on any reel. 1 × 5 × 5 + 5 × 1 × 5 + 5 × 5 × 1 = 75

Our simplified model thus contains 1 + 5 + 15 + 75 = 96 winning combinations. The table below shows the probability of a payout.

Calculate probability of slot machines
Winning combinationNumber of combinationsWinningReturns for 1 coinChance to win
3 jokers1303013.953%
Any fruit5105023.256%
2 jokers1546027.907%
1 joker7517534.884%
% for the winning combination44.444%Payouts99.537%

Calculation of payouts according to the formula

Σ (winning combination_k * possible yield_k) / (Total number of combinations)

(1 × 30 + 5 × 50 + 15 × 4 + 75 × 1)/(6 × 6 × 6) = 215/216 ≈ 0.99537

In this case, the slot machine has a payout ratio of 99.53%, which is very nice, but in a real casino, you will not find the same results. The average returns of slots online casinos will be between 94% and 98%.

The table also clearly shows how single coin wins affect payouts. If the win of each combination were equal to one coin, the winning ratio would drop to 44.4%. And that’s a very small number.

Analysis of a more complicated slot

Because the previous example was too distant from reality, let’s show you another example with higher numbers. To simplify, let’s assume again that there is only one payline, the slot machine has 3 reels and a total of 6 symbols that can appear on the panel:

SymbolReel 1Reel 2Reel 3

The total number of combinations is 23 x 23 x 23 = 12,167.

Winning combinations with single coin returns:

  1. 3x BAR, win 60 coins, number of combinations 1
  2. 3x SEVEN, win 40 coins, number of combinations 3 x 1 x 1 = 3
  3. 3x Cherry, win 20 coins, number of combinations 4 x 3 x 3 = 36
  4. 3x Other fruit, win 10 coins, number of combinations (5 x 6 x 6) x 3 = 540
  5. Cherry on two reels, win 4 coins, number of combinations 651
  6. Cherry on one reel, win 1 coin, number of winning combinations 3,880

Calculation for no. 5:

Cherry, Cherry, Other: 4 x 3 x (23 – 3) = 240

Cherry, Other, Cherry: 4 x (23 – 3) x 3 = 240

Other, Cherry, Cherry: (23 – 4) x 3 x 3 = 171

Calculation for no. 6:

First reel: 4 x 20 x 20 = 1,600

Second reel 19 x 3 x 20 = 1,120

Third reel 19 x 20 x 3 = 1,120

The following table shows the amount of payout and the chance of winning for the individual combinations.

Winning combinationNumber of combinationsWinningReturns for 1 coinChance to win
3x BAR160600.495%
3x SEVEN3401200.989%
3x Cherry36207205.934%
3x Other fruit540105,40044.507%
2x Cherry65131,93516.097%
1x Cherry3,88013,88031.979%
% of winning combinations42.007%Payout99.721%

As you can see, the payout ratio is very high again at 99.721% (12,133 / 12,161). If the slot were to pay a straight win for each winning combination in the amount of 1 coin, the payout ratio would be down to 42,007%.

So who pays for those Volcanic eruptions? Pirate Battles? Carnival Parades? and Glittering Showrooms? Slot Machines. 60-65% of casino revenue is generated by those bell-ringing one armed bandits that seem to multiply on casino floors like rabbits. So how does the average player gain an advantage and possibly win? Well... aside from cheating (which we really don't suggest you get involved in) the only way to gain some sort of advantage is to choose your slots with utmost care and discrimination.

Slot machines in Las Vegas are required by law to payout 75% of the money that goes into them, actual payout in Las Vegas is approximately 95%. Will you be the one that takes the money instead of gives it? That is up to luck, but with a little investigation one can easily learn to identify which machines are more favorable to the player than others. Slot machines are all about the payout... Red White and Blue, Double Diamond, Dick Fucking Clark, Cherry whatever. At the end of the day what every slot player needs to do is look at the pay schedule on the machine they want to play. Very often the same machine one row over will pay 5,000 credits on 3rd credit jackpot while you're playing on a 2,000 3rd credit machine. Plain and simple you're cheating yourself.

Slot Spotlights

A few notable slot machines we here at VT have found more playable, or more interesting, than the other nonsense out there like Leprachaun's Gold or Tabasco Slots or whatever.

We like machines that have the best payouts on the lowest winning spins. These will keep you going longer between larger wins and not enact the ATM-In-Reverse principle seen at many of the larger joints (Venetian being the worst we've experienced).

100 or Nothing
Red, White and Blue
Slot Machine Jackpots Photo Gallery
Wheel of Fortune
Wild Cherry

Slot Machine Payback Percentages

Below are the slot payback percentages for Nevada's fiscal year beginning July 1, 2002 and ending June 30, 2003:

5¢ Slot Machines
The Strip - 90.32%
Downtown - 91.50%
Boulder Strip - 93.03%
N. Las Vegas - 92.97%
25¢ Slot Machines
The Strip - 92.59%
Downtown - 94.83%
Boulder Strip - 96.47%
N. Las Vegas - 96.63%
$1 Slot Machines
The Strip - 94.67%
Downtown - 95.35%
Boulder Strip - 96.48%
N. Las Vegas - 97.21%

$1 Megabucks Machines
The Strip - 89.12%
Downtown - 88.55%
Boulder Strip - 87.76%
N. Las Vegas - 89.41%
$5 Slot Machines
The Strip - 95.33%
Downtown - 95.61%
Boulder Strip - 96.53%
N. Las Vegas - 96.50%
All Slot Machines
The Strip - 93.85%
Downtown - 94.32%
Boulder Strip - 95.34%
N. Las Vegas - 95.32%

The Math of Casino Slot Machines

For every dollar you wager in a slot machine, you will lose 100% - Payback% of that dollar. For example, you're at Bellagio playing the $1 Double Diamond slot, wagering Two Credits ($2) per spin. According to the table, for every $2 spin you will lose 5.33% of that bet... just shy of 11¢. Granted these 11 cents don't get extracted instantly... this is computed over time. So if your bank roll limit is $10 it will take you, on average 52 spins before your bankroll is toast (under $1) and you are out of credits.

Probability of a slot machine

How was this number derived:

STAKE x (Payback Percentages) = STAKE x (Payback Percentages) = STAKE x (Payback Percentages) = ...
Repeat calculation until the number gets below the minimum bet - if you play long enough, you're gonna go broke. THAT is a FACT. Slot machines are entertaining, relaxing, require little thought beyond pressing a button. IF you want to truly GAMBLE, you might want to look into Video Poker, and eventually Blackjack as other options.

Granted, sometime in there you just might hit that $500 win on the Wheel of Fortune, or The Elvis progressive might shake rattle and roll $1000 your way... but the math inside the machine determines that you will in fact lose a certain percentage of your wager on each spin, and the more you spin... the more you will lose despite short runs of successful jackpots. If you find you are UP... leave. Every spin of a slot machine generates a random number that has NOTHING to do with previous numbers. SLOT MACHINES do NOT run in streaks (even if you might wish to think they do). Don't expect to get any of the money you put into a machine out of it unless you learn to press the CASH OUT button.

Gamblers Library

VT fully recommends the following books to help you decrease the house advantage on casino card games.

Probability Of Slot Machine Formula

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