Probability Of Jackpot Slot Machine
- Appendices
- Slots Analysis
- Miscellaneous
Introduction
This page investigates the odds of the progressive jackpot slot machine, Megabucks, including the average jackpot and breakeven point.
For now, let's ignore the fact that a jackpot is paid by installments over 25 years and that the jackpot would be subject to income tax.
The probability of winning a significant amount of money on slot machines is slim to none. Casinos do not ordinarily disclose the odds of winning at their slot machines, so the slot players cannot be informed of their chances of winning. Believe me the chances of winning big money on the slots are very low. Slot machines odds for classic reel slot machines are basically defined by certain mechanical devices that control the work of reel-type slots. Each reel of the slot machine has a set of symbols imprinted on it. The chances of hitting a particular symbol are calculated through dividing it by the total number of symbols on each reel.
I don't know exactly how Megabucks is programmed. However, there is some information that is public knowledge. If we fit the pieces together, we can make a pretty good estimate of the point at which the return is 100%, known as the 'breakeven point.' Here is what we do know:
- According to John Robison, the probability of hitting the jackpot is 1 in 49,836,032. This figure comes from an article titled Megabucks closes in on record jackpot from the Las Vegas Sun, Dec. 24, 1999. That probability comes to (1/368)3, implying each reel has a 1 in 368 chance of stopping on the jackpot symbol.
- The Nevada Gaming Control Board indicates that the profit of Megabucks on both a percentage and dollar basis. The following is a summary for 1994 to 2009.
Megabucks Win — 1994 to 2009
Year Win ($) Win (%) 2009 53,352,000 10.43% 2008 83,981,000 11.85% 2007 88,858,000 12.72% 2006 100,923,000 12.39% 2005 100,923,000 12.39% 2004 67,326,000 10.54% 2003 83,069,000 10.41% 2002 76,842,000 11.98% 2001 69,821,000 11.50% 2000 69,103,000 9.75% 1999 74,921,000 12.28% 1998 134,943,000 12.25% 1997 66,166,000 12.18% 1996 57,619,000 10.03% 1995 65,223,000 10.48% 1994 46,760,000 9.44% total 1,239,830,000 11.39% The key piece of information from this table is that the overall profit of the game has been 11.39%. In other words, 88.61% is returned to the players.
- According to defunct source, starting in September 2005, Megabucks was reset to a jackpot of $10 million. Before that, the reset value was $7 million.
- According to a2zlasvegas.com, there have been 11 jackpots hit between September 2005 and the date of the last jackpot (Feb. 21, 2010). That same website shows a jackpot was hit on September 15, 2005. The number of days between then and the time of this writing is 1,619 days. We also see from that website that the total of the last 11 jackpots was $167,367,727. Of that, $110,000,000 was from the reset amounts and $57,367,727 was from the progressive contribution.
Probability Of Winning Slot Machine Jackpot
We can estimate the number of times Megabucks was played during the 1619 day period by dividing the number of jackpots of 11 by the probability of winning: 11/(1/368)3 = 548,196,352. Assuming each player bet the $3 required to win the jackpot, then a total of $1,644,589,056 was bet.The portion of money returned to players in form of jackpots is thus $167,367,727/$1,644,589,056 = 10.18%. From the Nevada Gaming reports, we know a total of 88.61% is returned to players. That means that the portion returned to players in non-jackpots is 88.61% - 10.18% = 78.44% (The 0.01% apparent difference is due to rounding).
If there were no small wins, and no progressive contribution, then the return of the game would be $10 million/(3×(1/368)3) = 6.69%. As already shown, the total return from jackpots is 10.18%, leaving 3.49% coming from the jackpot meter. Here is a summary of where each $1 bet on Megabucks goes:
Megabucks Breakdown
| Item | Cents |
|---|---|
| Fixed wins | 78.44¢ |
| Meter reset | 6.69¢ |
| Progressive contribution | 3.49¢ |
| Profit | 11.39¢ |
| Total | 100.00¢ |
The average point at which the jackpot will hit is 10 million + [$3 × 0.0349 / (1/368)3] = $15,215,248. In 2006, when the jackpot was almost $16 million, IGT, Megabuck's creator, purchased ads in the local media stating that the jackpot was 'overdue' to hit. I'm quoted in a Las Vegas Sun article about it, titled 'Pennies ready to pop'. This would seem to indicate my $15.2 million figure is not far off.
If j is the jackpot at which the game becomes a fair bet, with a 100% return, then we can solve for j as follows:
Probability Of Jackpot Slot Machines
1 = 0.7844 + j × (1/368)3/3
j × (1/368)3/3 = 1 - 0.7844
j = 3 × (1 - 0.7844) / (1/368)3
j = $32,238,319.
The probability of any given jackpot growing this big is 1.41%. At the current rate of play, a jackpot should get this big once every 29 years, on average.
At any given time the return can be estimated as 78.44% + 0.6689%×m, where m is the number in millions of the current jackpot. For example, at a jackpot of $15 million, the return would be 78.44% + 0.006689×15 = 88.47%.
Everything in this page should be taken as a ballpark estimate. Various factors could cause it to be off, including players not betting the full $3 and the fact that while 11 jackpots were hit in the study period, the expected number could be higher or lower.
It also bears repeating that the above does not factor in the annuity or taxes. Let's look at what happens if we do consider those factors. For the time value of money, let's use the return on long-term Treasury Bills. Megabucks jackpots are paid in a 25-year annuity. At the time of this writing a 20-year T-Bill paid 4.58% interest, and a 30-year one paid 4.74%. Let's split the difference at 4.66%. Using some actuarial math I won't get into, the value of the annuity is worth 61.07% of face value, based on that interest rate, and 25 annual installments, at the beginning of each year.
For taxes, let's assume close to the expected jackpot of $15 million. Under 2010 income tax rates, assuming the winner is filing jointly, and all other income exactly equals deductions, the taxes due will be 30.05% for 2010. Assuming no change in the tax law, that will drop over time, because the tax brackets will be adjusted upward, but the winning payments won't be. I tend to think the recent passage of health care will increase tax rates, especially on large incomes. Let's just assume those factors cancel each other out, to keep it simple.
So to keep things in round numbers, the winner will keep 61% after the annuity, and 70% of that after taxes. So the jackpot winner will see about 61% × 70% = 42.7% of his winnings in current dollars. Factoring the annuity and taxes, the breakeven point becomes $75.5 million. The probability of any given jackpot growing that big is about 1 in 283,000, and will happen once every 114,000 years. Again, I'm making lots of assumptions, so these estimates should be considered very rough.
After publishing this article, a reader quoted a page at slot-machine-resource.com, which states that after the first installment is made, the player is given the option to get 60% of the rest immediately, or stick with the installment plan. Tax implications aside, which favor the annuity, the interest rate at which the two options are equal is 4.581%.
External Links
- Megabucks Closes in on Record Jackpot from the Las Vegas Sun, Dec. 24, 1999.
- Nevada Gaming Control Board
- Slots Payout percentage, from Cassaon Casino.
- History of Megabucks Jackpots, from a2zlasvegas.com.
- Pennies Ready to Pop, from the Aug. 9, 2006 Las Vegas Sun.
- Megabucks, from slot-machine-resource.com.
Written by: Michael Shackleford
Remember the movie National Lampoon’s Vegas Vacation, when gambling fever consumes Chevy Chase’s character, Clark W. Griswold? He goes on a losing streak to beat all losing streaks while his son, Rusty, wins four cars by playing the slot machines. Maybe Clark would have done better if he had read Probability For Dummies! In this article, you discover the basic ideas behind slot machines and how they work, so that you can get past the myths and develop a strategy based on sound probability.
Understanding average payout
When casinos advertise that their slot machines pay out an average of 90 percent, the fine print they don’t want you to read says that you lose 10 cents from each dollar you put into the machines in the long term. (In probability terms, this advertisement means that your expected winnings are minus 10 cents on every dollar you spend every time the money goes through the machines.)
Suppose you start with $100 and bet a dollar at a time, for example. After inserting all $100 into the slot, 100 pulls later you’ll end up on average with $90, because you lose 10 percent of your money. If you run the $90 back through the machine, you’ll end up with 90 percent of it back, which is 0.90 x 90 = $81. If you run that amount through in 81 pulls, you’ll have $72.90 afterward (0.90 x 81 = 72.90). If you keep going for 44 rounds, on average, the money will be gone, unless you have the luck of Rusty Griswold!
How many pulls on the machine does your $100 give you at this rate? Each time you have less money to run through the machine, so you have fewer pulls left. If you insert $1 at a time, you can expect 972 total pulls in the long term with these average payouts (that’s the total pulls in 44 rounds). But keep in mind that casinos are designing slot machines to go faster and faster between spins. Some are even doing away with the handles and tokens by using digital readouts on gaming cards that you put into the machines. The faster machines can play up to 25 spins per hour, and 972 spins divided by 25 spins per minute is 38.88 minutes. You don’t have a very long time to enjoy your $100 before it’s gone!
The worst part? Casinos often advertise that their “average payouts” are even as high as 95 percent. But beware: That number applies only to certain machines, and the casinos don’t rush to tell you which ones. You really need to read or ask about the fine print before playing. You can also try to check the information on the machine to see if it lists its payouts. (Don’t expect this information to be front and center.)
Implementing a simple strategy for slots
Advice varies regarding whether you should play nickel, quarter, or dollar slot machines and whether you should max out the number of coins you bet or not (you usually get to choose between one and five coins to bet on a standard slot machine). In this section, you’ll find a few tips for getting the most bang for your buck (or nickel) when playing slot machines.
Basically, when it comes to slot machines, strategy boils down to this: Know the rules, your probability of winning, and the expected payouts; dispel any myths; and quit while you’re ahead. If you win $100, cash out $50 and play with the rest, for example. After you lose a certain amount (determined by you in advance), don’t hesitate to quit. Go to the all-you-can-eat buffet and try your luck with the casino food; odds are it’s pretty good!
Choosing among nickel, quarter, and dollar machines
The machines that have the higher denominations usually give the best payouts. So, between the nickel and quarter slots, for example, the quarter slots generally give better payouts. However, you run the risk of getting in way over your head in a hurry, so don’t bet more than you can afford to lose. The bottom line: Always choose a level that you have fun playing at and that allows you to play for your full set time limit.
Deciding how many coins to play at a time
When deciding on the number of coins you should play per spin, keep in mind that more is sometimes better. If the slot machine gives you more than two times the payout when you put in two times the number of coins, for example, you should max it out instead of playing single coins because you increase your chances of winning a bigger pot, and the expected value is higher. If the machine just gives you k times the payout for k coins, it doesn’t matter if you use the maximum number of coins. You may as well play one at a time until you can make some money and leave so your money lasts a little longer.
For example, say a quarter machine pays 10 credits for the outcome 777 when you play only a single quarter, but if you play two quarters, it gives you 25 credits for the same outcome. And if you play the maximum number of quarters (say, four), a 777 results in 1,000 credits. You can see that playing four quarters at a time gives you a better chance of winning a bigger pot in the long run (if you win, that is) compared to playing a single quarter at a time for four consecutive tries.
The latest slot machine sweeping the nation is the so-called “penny slot machine.” Although it professes to require only a penny for a spin, you get this rate only if you want to bet one penny at a time. The machines entice you to bet way more than one penny at a time; in fact, on some machines, you can bet more than 1,000 coins (called lines) on each spin — $10 a shot here, folks. Because these machines take any denomination of paper bill, as well as credit cards, your money can go faster on penny machines than on dollar machines because you can quickly lose track of your spendings. Pinching pennies may not be worth it after all!