A simple probability question on the odds of getting a flush

Let’s assume you are playing a $1 machine and inserting $5. The machine pays the standard 9 times your bet for a full house and 6 times your bet for a flush.

You are dealt: 2c 3c 7c Jc Jd

If you hold all five cards, you get back $5 every time for your pair of Jacks or better.

If you discard the Jd and draw to the flush, for each 47 trials you will make Nothing 36 times, pair of Jacks or better 2 times, and a flush 9 times, for an average payback of $5.9574.

If you hold the pair of Jacks, for each 16215 trials you will make a pair of Jacks or better 11559 times, Two pair 2592 times, Three of a kind 1854 times, Full house 165 times, Four of a kind 45 times, for an average payback of $7.6827.

So clearly, with a four flush and a pair of Jacks or better, it is better to draw to the pair than to the flush.

These calculations were made with the demo version of WinPoker

Oh crap. Misclicked and posted before I was finished.

WinPoker. I played video poker for a living for a couple of years and this is the software I used. The website also has a link to VP Strategy Master, which will generate the best strategy for different games and pay schedules.

If I were playing today, I would get Video Poker for Winners!, which does all that the other two do, and more. Bob Dancer is the authority on video poker play. Check his website for his free lesson schedule if you’re going to Vegas; very worthwhile.

If you want to understand video poker, the book to read is Video Poker-Optimum Play.

Now I’m done. :slight_smile:

Wait. If you hold the pair, then shouldn’t you always make a pair of Jacks or better?

In video poker terminology a “pair of Jacks or better” means specifically: JJ or QQ or KK or AA.

IOW, one high pair and one high pair only.

Yes, you will always have a pair of jacks. At least. But sometimes you will get a *better *hand than a pair of jacks.

11559 is the number of times you will only have a pair of jacks.
2592 times, you will have a pair of jacks, and another pair
1184 times you will have three jacks,
165 times you will have a full house, with either two or three jacks
45 times you will have 4 jacks

is that clear?

Ah, I see. I was parsing it as “(a pair of jacks) or better”, when I should have been parsing it as “a pair of (jacks or better)”.

I actually had used some form of “parse” in my original response and changed it. You just went up a notch in my book. :smiley:

What machines are these that are so horribly planned that this is actually possible? Surely no casino would allow the mathematical possibility of a payout percentage above 100%?

The original video poker machines in Las Vegas got a lot play when they were first introduced but they stopped getting play very quickly. The first machines did not pay back on Jacks or better and the locals soon decided they were not worth playing. The manufacturers changed the pay schedule to return the bet on a pair of Jacks or better and the play took off. The payback on what is now known as “Full Pay Jacks or better” is 99.54% with optimal play.

You can still find some of the original machines around Vegas. Look for old style CRT displays. The original machines have a red plastic plaque glued on the front that says “Bet returned on pair of Jack or better.” Slightly newer but still CRT machines list the Jacks or better payout on the glass. Modern machines have all the payouts listed on the flat screen display.

In the early days of the game, there were people (like me) writing BASIC programs on our 8K Apple computers to analyze the game and determine the best strategies. The manufacturers were pretty much guessing at the proper payouts and they made some mistakes.

The machines now can all be adjusted to different payout schedules. Some of them are beatable right off the top, but you mostly only find machines with loose settings in places with a lot of competition. Where I live, in PA, the casinos are all around an hour drive apart … so they basically each have “the only game in town” and run most of their machines on the tightest setting. There are, however, Full Pay Jacks or better machines, but they don’t offer cash back bonuses, just comps, in which I am not interested.

There are a lot of variations of video poker. Not all are beatable, but many are. Some are much easier to beat than others. With a Full Pay Deuces Wild machine, any reasonably bright person can learn to play with maximum advantage with about 10 hours of study if they are starting from scratch on their own. If they have an experienced player to teach them, they can be playing with an advantage in 10 to 15 minutes.

Even if a machine is not set to be beatable by optimal play, they can often be beaten when casino promotions are included. Most casinos these days offer some sort of cash back or comps for slot play and in areas where they have competition those offers can be substantial.

A common deal is Vegas is to give slot players 0.5% cash back for their play – and promotions are frequently offered for double or even triple point days. Add the 99.54% payback for Jacks or better and 0.5% for the slot club and you are playing at 100.04% – on double or triple point days, add another 0.5% or 1%. Deuces Wild pays back 100.76% – add promotion bonuses to that and you can do quite well.

If playing Jacks or better at 100.54% on a double points day doesn’t sound tempting, let’s take a look at what that actually means. It’s often shocking to people when they first realize how much money they are pumping through a slot machine.

You insert $5 into a $1 machine on each play. Typical casual players will play maybe 500 to 600 hands per hour, pros will hit over 700 and maybe reach 1000 if the machine is set for fast play. We’ll use 700 as a sustainable speed. $5 times 700 hands per hour = $3500 per hour times 100.54% payback = $18.90 per hour profit. That’s substantially better than minimum wage and we are playing a game that has a negative expectation. There are also $5 machines and $25 machines and $100 machines and there are triple point days and there are machines that have a better payoff schedule. See where this might lead?

On top of the cash back bonuses, casinos also often send out coupons to regular players that can be turned into cash just by going to the casino. There was a place called The Reserve off the strip in Vegas (in Henderson, actually) that had a 50 cent Deuces Wild machine that would deal up to 1200 hands per hour. I couldn’t sustain that speed, but for the speed I played, triple point day was worth a little over $56 per hour (yes, it is possible to calculate your expectation very precisely in video poker) … and on top of that, every week they sent me a coupon that I cashed in for $120 … and on top of that, they had drawings for substantial amounts of cash to regular players … and on top of that, I got comps to their pretty decent restaurants … and I could have gotten a free room if I wanted it. And that was just for spending one day a week playing a 50 cent video poker machine.

So yeah, such machines exist, mostly in areas where the casinos have competition but opportunities can pop up anywhere there are games. Casinos make mistakes; personality and schmoozing ability are much more important factors in becoming a Casino Manager than actual deep knowledge of the games – and I don’t mean anything nasty by that – if I owned a casino, I would hire the schmoozer over the egghead geek (which I am) too.

RickJay, you are asking a lot of questions that can lead a bright person off the path of the clean cut 9 to 5 lifestyle and into the dark-of-night recesses that most people would do best to avoid. :wink:

One more thing: Before the poker explosion brought on by the internet and TV there were around 200 people in Vegas who actually earned their living playing poker – there were many more than that who earned their living playing slot machines. To this day, most people, including most professional poker players aren’t aware of that.

Fascinating. But couldn’t a casino, in theory and likely in practice, exercise its right to deny its gaming to someone who appeared to have found a way to exploit the system to a profitable advantage? You’d think they’d notice.

Certainly, they have a reputation for being less than hospitable to anyone who demonstrates an ability to card-count in blackjack.

That’s another complicated question – they could but they don’t. Most casino management people don’t really think the machines can be beat and the few who do don’t think enough people would actually want to do it to matter, and on and on …

Million Dollar Video Poker goes into a lot that. It’s not about how to beat video poker, it’s about how Bob Dancer beat video poker. He writes a lot about the tricks and maneuvers of disguising the fact that you’re a winning player and about how he handled it when a casino did decide to ban his play.

Casinos have been hiring him to give free Video Poker Classes for years. The classes are the real thing, not just the Get Rich in Real Estate type free sessions trying to sell you something. Not many people follow through after taking the class but the information he provides is the real thing.

  1. Do you mean normal slot machines or video poker? If the former, I am among those who don’t buy it. If a person could sit at one machine 24/7, they could probably hit a few jackpots once in a while. And if they could do it for weeks at a time, they might hit a very large one. But I can’t imagine a casino putting in a machine that sends out more money than it takes in. What would be the point? To entice players? If someone found that machine, everyone would be lining up to play it.

  2. In your calculations regarding video poker, isn’t it true that the odds of “beating” the machine always take into account the fact that you must play maximum bet in order to do so? The real edge in maximum betting comes in hitting the Royal. Otherwise, the proportions spent and received are the same. You have to have one really out of proportion payout to arrive at winning odds, true?

Yes the point is to entice players. Casinos will often advertise - with up to 100.74 (or it might be 100.47)% back.

Three things:

  1. Basic strategy on these is very complicated - most people don’t know it - it is first come first serve - and most likely the AVERAGE payoff is less than 100%.

  2. They get enough people who come in - find those machines are busy - but play something else.

  3. Usually it is something like 7 machines - and only quarters. Even if EVERYONE played perfect basic strategy it isn’t like it would bankrupt the casino. You can only play so many hands an hour (the animation speed of the cards being dealt is kind of the limiting factor)

It is sort of like a loss leader - where a store might sell 5 iPads below their costs.

Both. While we’re talking mainly about video poker, there are other types of machines that are beatable.

Wild Cherry Pie slot machines have the normal spinning reels and they also have a sort of side game; there is a graphic of a cherry pie at the top of the machine. When the reels spin, on random occasions one of the cherries gets filled in. When the last cherry in the pie gets filled the machine pays out a small jackpot.

The first Wild Cherry Pies were $5 machines put in the Tropicana in Las Vegas. Lots of people would play for a while and leave the machine, leaving the pie partly (or almost) full. The Pie jackpot could be won by inserting only one coin; not necessary to play max coins. So some observant people began simply waiting for players to leave a machine on which the pie was nearly full and then playing one coin at a time until the mini-jackpot hit. They would then cash out and look for another machine that was ready to hit.

Some greedy ones started actively and aggressively doing things like blowing smoke on or coughing on players to cause them to leave machines that were ready to pay off the mini-jackpot. They bribed security guards when the customers complained, sometimes actually causing the legitimate customers to be thrown out of the hotel. They were making around $2,000 a day.

The $5 machines only lasted a few months but $1 machines started popping up all over town. There were always people standing behind them, waiting for a player to leave a nearly filled pie. With no skill and almost no risk, they were making $300 to $400 a day. I only know of one person who ever had a losing day at it.

So yeah, there are opportunities and I personally know people who have been earning their living playing slots for over 30 years. Whether you buy it or not.

Have I understood, though, that you have to play a long time in order to make a profit?

AIUI, you will lose steadily, until you get a Royal Flush with five coins. Then you win the jackpot which recoups your losses, and more. And since that happens only once every 40,000 hands (assuming no wild cards) you’ll have to play around 57 hours on average before you’re in profit.

Is that right?

Ever heard of a loss leader? Basically, a product or service that businesses sell at a loss, but which attracts customers to more profitable lines.

A casino doesn’t mind someone winning $18 per hour on VP as long as there are other people losing $180 per hour on roulette.

Plus, if you don’t play optimum strategy, you’ll lose.

You got it. On Jacks or better you expect to lose at about 2.5% or nearly $90 per hour on a $1 machine. You get it back plus your profit when you hit the royal.

It’s a long-term proposition. By the time you reach about 200,000 hands you will be very close to where you mathematically expect to be. If you play around with that demo of WinPoker I linked to earlier you can see how precise it is possible to be with video poker calculations. Set it to Training Mode and you will find it popping up saying things like “Major Error!!! Cost: 25.22 cents” when you make a mistake.