Buy More Tickets

(*Note: this column was written on the 3rd day of the recently completed government shutdown)

            As we finish Day 3 of the government shutdown, it is hard to find a news story that is about anything else.  I was pleasantly surprised (initially) when my eye caught a story on Yahoo's main news page that dealt with lotteries.  Apparently, they were revealing the secret of how to win the Powerball Lotto.  Who doesn't want to win Powerball?  I read the article, which was all of about two paragraphs. 

            It began by talking about last month's sole winner, who won $400 million.  He 'beat the odds' by hitting the lottery which is a 1 in 175 million chance (I'll take their word for this).  Per the article, the odds of getting struck by lightning is a mere 1 in 10,000 (again, I'll take their word for this).  Finally, the article gets to the important question "How can you increase your chances of winning the lottery?"  Then they apparently provide the way.  A statistician in Louisiana has discovered that certain numbers come up more often than others!

            The most frequent powerball number is 20.  The most common white ball numbers are 42, 16, 35, 26 and 19.  There you have it.  The winning numbers!  So, to increase your changes all you need to do is play these 5 numbers with the number 20 and you can start spending your millions!   I didn't actually look up the historical winning numbers, but I'm going to take a strong guess that this particular combination has NEVER come up before, but undoubtedly they are on their way.

            Now, nothing in this article gives the actual frequencies of these numbers.  Nothing shows that they show up an abnormally high amount of times.  Let's assume that Powerball has been drawing twice a week for 20 years.  That is 2000 total draws.  That is 10000 numbers drawn (white balls) and 2000 red balls drawn.  There are 59 white numbers and 35 red balls.  On average each white number should be drawn 169.4915 times.  So, for all we know the 5 numbers he cited showed up 170 or 171 times while the rest of the numbers showed up 169 times.  Clearly a massive statistical edge!   On the red balls, the average is 57.14.  So, 20 may have shown up a couple more times that all the others.  Again, a clear statistical advantage!

            I've always been strong at math.  I realize not everyone is.  I don't expect everyone to be.  But, bad math packaged as an article on the front page of Yahoo news really drives me absolutely nuts.  To add insult to injury, the article went on to suggest you should buy your ticket in Pennsylvania because that's where the largest number of winning Powerball tickets have been sold.  It doesn't even take into account the possibility that more tickets have been SOLD in that state than many others! 

            I've often said (semi-jokingly) that the world's largest casino is the stock market.  But, there is one critical way that the stock market greatly differs from gambling.  With stocks, past performance CAN BE used to determine the likelihood of future performance.  While there are no guarantees, there are likelihoods.  A stock that has paid a dividend for the past 100 years is not likely to stop paying it next year (barring any specific news being known).  Stocks have their ups and downs, but you are NOT really dealing with random events.

            The same cannot be said for what happens in a casino (or a lottery).  The last 3 numbers on the roulette wheel could've been red and the likelihood of the next number being red will still be 18/37 (or 38).  The last 10 numbers could've been red and the probability will STILL be 18/37 (or 38).  The last 5 hands of video poker could've contained the 2 of diamonds in the initial deal and the probability of it showing in the next hand's deal will still be in 5 in 52.   Nothing changes when we are talking about the lottery.  It does not matter if one number has appeared more often than others.  Next week's numbers are completely random and each number has the same probability of being drawn as the next. 

            Okay, I'll admit it.  Maybe I'm just jealous that after a decade of writing for Gaming Today, Yahoo has not covered a single one of my columns, but some guy writes a complete nonsensical piece of useless information and that makes their front page.  The sad part will be how many thousands of people will read that article and actually run out to play those numbers.  There is only ONE way to increase the likelihood of winning the lottery - buy more tickets. 

What is the Allure of Progressives

            There is a theory in physics that goes for every action there is an equal and opposite reaction.  In gambling, there is a similar theory.  For every table game there will eventually be a sidebet.  And, for every sidebet there will be a Progressive version of the sidebet.  The math behind Progressives is probably the least understood math of any type of gambling.  It really isn't that hard once it is explained properly, but I've worked with a lot of inventors on a lot of Progressives, and it is fairly obvious to me that few people, even in the industry, understand how a Progressive works mathematically.

            Generally speaking there are 3 components of a sidebet - the fixed pays, the seed and the contribution rate.  Normally when we calculate the payback of a sidebet, we simply multiply the fixed pays by the frequency of each winning hand and sum up these values.  For a Progressive, we have to alter one step slightly and add one.  For the jackpot event, we use the seed amount as the equivalent of the fixed pay for that event.  Each time it is hit, the casino is on the hook to put that money back on the meter, so it is similar to a fixed pay in that regard.  We then need to add the contribution rate - which is the amount of each dollar wagered that goes on the meter - to the total payback calculated.    I'll save more details for another day, as this is not the point of today's column.  What is the point is to discuss how a Progressive differs from other wagers.

            While the top pay for most sidebets are pretty large, the amount they contribute to the overall payback is usually pretty small.  If you pay 1000 for a 1 in a million even, the contribution rate is a meager 0.1%.   In video poker the Royal Flush contributes only 2% to the payback of the game.  If we were to look at most table game sidebets, we'd probably find that most top pays contribute about 1-2% (or less) to the overall payback.  But, when we switch to a Progressive, we find that the top pay frequently contributes 15-20% to the payback when we take into account both the seed and contribution rate.  What does this mean for the Player?

            As I said, the Royal Flush accounts for 2% of the payback of video poker.  What this usually means is that until you hit one, you're only playing at about 97.5% which can be a bit rough.  When you hit one - and if you are a regular player, you WILL hit one, you bring the theoretical payback back to 99.5%.  Hit the Royal more frequently than 'normal' and you're likely up money as you will be above 100%.  With Progressives, it doesn't quite work the same way.  That top hand is either more rare or you'll be playing a game that deals much more slowly than video poker, meaning that there are no guarantees that you will EVER hit it.  So, even if the sidebet were paying 99.5% like video poker, ONE PLAYER is going to wind up winning 15-20% of that payback and everyone else will be playing at 77.5% - 82.5%. 

            When you consider the fact that many sidebets have paybacks far lower than 99.5%, you realize that the picture for those that don't hit the jackpot is even more bleak.  So, why do people play Progressives?  There are two main reasons.  One is a bit emotional and the other a bit more practical.

            First, Players have always been willing to accept low paybacks for a chance to win a life-changing amount of money.  The Lotto has made a lot of money for a lot of states.  Most states payout only 70% on their lotteries.  This is lower than the legal minimum of any casino game here in Nevada.  But, for the chance, however slim, of winning millions of dollars, Players are willing to throw a few dollars in for the hope of getting struck by lightning. 

            The second reason deals with the way Progressives work and makes far more mathematical sense.  To the casinos, the payback of a game is the long-term payback, which is calculated as I described earlier.  You'll note that what I described completely ignores the specific value on the meter at any point in time.   This money is merely an accumulation of the contribution rate over time.  It really doesn't matter to them (mathematically), if a jackpot that is supposed to hit about once a year, doesn't get hit for 3 years.  However, to the Player, the payback of ANY wager is dependent upon the specific payouts for each winning hand at the point in which you make the wager.  It doesn't really matter if the contribution rate is 10% of 20%.  If a Jackpot which is supposed to average $250,000, goes all the way up to $600,000 then the payback at that point in time is WELL above the theoretical payback. 

            It is possible that at a particular point in time that the payback of a wager could be over 100%.  At this point, it makes sense to play the game mathematically.  The problem is, however, that it will be one person that will benefit from this occurrence and it may not be you.  Then again, it might!

AMERICA BEATS ITSELF INTO A LOTTERY FRENZY OVER MEGA MILLIONS

            As I'm writing this, the country is beating itself into a frenzy not over politics but over a lottery.  The Mega Millions Lotto has an estimated prize of $640 MILLION.  That would make it the largest jackpot in the world.  Lotteries tend to have paybacks of about 50-60% so they aren't exactly a wise wager.  Yet, as I have often written, people are more willing to wager in games with bad paybacks if the top prize is life altering.  I think more than half a billion dollars meets that requirement.  I have to admit that if Nevada participated in Mega Millions, I would've tried to get some tickets.  I was NOT motivated enough to drive to nearby California to get them, however.

            Even when your choice of game is something like a Lotto, I think you should go in with your eyes open.  The odds of winning the top prize is about 176 million to 1.  To put that into a casino perspective, that is a little higher than the odds of being dealt a sequential Royal (10-A or A-10) in SPADES on the deal in video poker!  Of course, even if you're playing a Reversible Royals video poker machine, you're only going to get paid maybe $40,000 for that hit, not $640 million.

            Unfortunately, unlike most casino games, it is a bit more difficult to determine the expected value of this week's drawing for one major reason.  The $640 million dollars will be SPLIT by each of the people who have a winning ticket.  The lottery has stated that $1.5 billion worth of tickets have been sold, but from reading further it would appear that this is the TOTAL number of tickets sold since the last time the jackpot was won.  This does NOT represent the number of tickets sold for this particular drawing which is all that matters.  If we actually knew how many tickets were sold for this drawing, we could determine a more accurate expected value. 

            Armed with this information - and $176 million, it might actually pay to buy every possible combination of numbers.  We would then actually be wagering on how many other people hit the same set of numbers.  If less than 3 others, we would actually make some money on the deal.  Well, BEFORE Uncle Same takes hit cut anyhow.  To really make money, we'd probably have to be the only one to have the winning ticket.  History tells us this is unlikely and even less so if you were to add in someone who bought EVERY ticket. 

            No one plays these types of lotteries believing it is a wise investment.  We all know that the odds are very long.  The payback of the lottery is normally around 50-60% and even when it grows this large, it is probably no more than 70-80% when we consider that we are likely going to have to share it should we get struck by lightning and actually win.  What I find most amusing about these situations is the comments we get from some people.

            Today, I was reading a rather whimsical article about just how much money the $1.5 billion that was spent on lottery tickets really is.  It talked about how many families it could feed and how many trips to the space station you could make with this type of money.  Sadly, it also explained how it was only 0.1% of the national debt.  The article then moved on to quote some people who chose to play and why.  Of all the things I read, the one that made me to a double take came from an accountant in Louisiana (I won't post his name here):

            The article stated that the gentleman had bought 55 tickets and that he knows buying that many tickets doesn't mathematically increase his odds, and that his $55 could have gone elsewhere. He spent it anyway.

            "Mathematically, it doesn't make a difference, and intellectually we know that. But for some reason buying more tickets makes you feel more lucky," the accountant said. "Even people who know better are apt to feel that way."

            Maybe, he bought 55 tickets all with the same numbers?  Mathematically, buying more tickets doesn't make a difference?  So, if I buy one ticket I have the same chance to win as someone who buys 2 tickets?  What about the guy who buys 10?  or 50?  or 55?  or 176 million?  As an accountant, I would think he would understand numbers a bit better.   If you buy 2 tickets your probability of winning doubles as compared to buying 1 ticket.  If you buy 10 tickets your probability of winning multiplies ten-fold.  This gentleman bought 55 tickets, so he brought his odds down to a mere 3.2 million to 1 of hitting the big jackpot.  

            Fortunately, playing the Lotto takes as much skill as playing slot machines.  But many of the same people walk into a casino armed with about the same level of knowledge of the games.   Yes, mathematically, we know in the long run that we are likely to lose, but that doesn't mean we should take prudent steps to keep our losses to a minimum and give ourselves the best chance to win in the short run.  Because, in the end, mathematically, it all makes a difference.