On November 28, 2012, the Powerball jackpot reached a record $590 million. The jackpot had rolled over several times. The question becomes: What are the odds that at least one ticket will win the Powerball jackpot relative to the amount of prize money?
The amount of prize money is usually 50% of total ticket sales. If we know the price per ticket we can determine total number of tickets played. The Powerball ticket costs now $2. Thus, total sales were 2 * 590,000,000 (as the prize pool represents around half of total sales). However, one ticket costs $2, so we approximate the number of tickets sold to 590,000,000.
What correct method do we apply to calculate the odds that at least one ticket will hit the Powerball jackpot?
One quick tendency would be to apply the Birthday Paradox or the probability of coincidences. That's the wrong path to the correct answer.
We can correctly apply the Fundamental Formula of Gambling (FFG):
DC represents the degree of certainty that an event of probability p will appear within a number of trials N.
DC is the essential element here. We calculate the degree of certainty from FFG by using my probability software SuperFormula (component of the scientific software package Scientia). We have the following elements:
p = 1 / 175,223,510
N = 590,000,000
Doing the calculations, we get the most accurate result:
DC = 96.5%.
That was the chance that at least one ticket would hold the Powerball jackpot combination for the above amount of prize money or total tickets sold.
Correct interpretation: In the total number of tickets sold, only 96.5% of total possible Powerball combinations were played. The rest, or 3.5% of combinations, were NOT played (because of repeat-combinations).
The previous Powerball jackpot was set at some $450,000,000. The degree of certainty that at least one ticket would hold the Powerball jackpot was: DC = 92.3%.
The two previous calculations are closely related to Ion Saliu's Paradox of N Trials. If p = 1 / N and we perform N trials (e.g. randomly generating N combinations similar to playing randomly N tickets), the degree of certainty DC tends to 1 1 / e (or 63.2%) when N tends to infinity. A group of Australian investors played the Virginia lottery when the lotto jackpot reached over $100 million (in the 1980s or 1990s).
The investors played ALL combinations in that particular lotto game or so they thought! In fact, they played quick picks or randomly generated combosnations by the lottery computer. The random picks played by the investors covered only around 63.2% of all possible lotto combinations in that particular game (fewer than 14 million)! Nevertheless, they won the jackpot!
I heard on TV the same November 28, 2012, that they heard of Ion Saliu's Paradox of N Trials but they got it wrong. They were saying that the chance for a ticket to win that 590-million-dollar Powerball jackpot was 60%. NOT!
This paradox strongly indicates that it is better to play more tickets at once, instead one ticket over a longer period of time. The chance is better to play 100 tickets in one lottery drawing than playing 1 ticket in 100 draws!
We can look at the problem from a totally different angle. How big should the Powerball jackpot be to have a 50% degree of certainty (DC, or chance to a layperson) that one ticket will win the jackpot?
We can put to work the same great SuperFormula program; function: F = Fundamental Formula of Gambling (in fact, number of trials N). Parameters:
p = 1 / 175223510
DC = 50%.
Result: N = 121,455,681 tickets; that is, a jackpot of some 120 million dollars.
Interpretation In around half the cases (50%) when the Powerball jackpot reaches 120 million dollars, we should see a rollover once; we should see a winning ticket once. Perhaps in real-life there are more Powerball jackpots under $120 million dollars. It is so because even the rudimentary random number generators of the state lotteries do not generate combinations defying standard deviation. I doubt they generate combinations such as:
1 2 3 4 5, 1
55 56 57 58 59; 35.
Thus, a pretty large amount of Powerball combinations are left out. Also, many lottery players play birthday numbers evidently, from 1 to 31. This huge-jackpot Powerball drawing of November 28, 2012, consisted of birthday numbers only: 5 16 22 23 29, 6. It happens more often than I would expect.
Those are just two of the reasons why the Powerball jackpots do not fly above the 120-million-dollar horizon more frequently.
You know what? I did not play. I put money aside and I waited for the next drawing. I had expected the jackpot would reach three quarters of a billion dollars ($750 million)! Didn't I tell you about Ion Saliu's Paradox?! The chance to encounter at least one winning ticket for that $750 million Powerball jackpot would have been: DC = 98.6%.
And, don't forget, there is NO absolute certainty, or DC = 100%! Absolute certainty is absolute absurdity in mathematics.
p = 1 / 292,201,338
N = 2,200,000,000
Doing the calculations in SuperFormula, we get the most accurate result:
DC = 99.95%.
The largest Mega Millions jackpot was recorded on March 30, 2012: 656 million dollars. The amount of prize money being 50% of total sales, it must be that total amount of tickets sold was 1,312,000,000 (1.3 billion tickets!)
The new parameters are:
p = 1 / 175,711,536 (the odds to win the Mega Millions jackie)
N = 1,312,000,000 (number of trials or number of tickets sold)
Doing the calculations, we get this most accurate result:
DC = 99.9%.
That was the chance that at least one ticket would hold the Mega Millions jackpot combination for the above amount of prize money or total tickets sold.
Correct interpretation: In the total number of tickets sold, 99.9% of total possible Mega Millions combinations were covered. Chances are, even Mega Millions combinations such as 1 2 3 4 5, 1 or 52 53 54 55 56; 46 were possibly played.
When a Mega Millions jackbarrel reaches $100 million, it means some 200 million tickets were sold. The new parameters are:
p = 1 / 175,711,536
N = 200,000,000
DC = 68%.
That's the chance that at least one ticket will hit the Mega Million jackpot when it reaches $100,000,000.
It can be undeniably verified by correct software and fast computers. I wrote the software: OccupancySaliuParadox. It tackles also the famous Classical Occupancy Problem. Granted, the software works only with 10 digits (0 to 9) because of speed considerations. My lottery software generates 1000 random pick-3 sets (from 000 to 999). I get consistently around 630 unique sets, as calculated by the FFG. Easy to do it 100 times and purge the duplicates (repeats). The average of all unique figures comes very close to the theoretical value.
It would take a very long time to generate all Powerball or Mega Millions combinations, and then clean up the repeat-combinations and then repeat the process at least 10 times! But the mathematical foundation is the same as for the 10-digit case. One can never randomly generate and cover all combinations with a 100% degree of certainty. The logarithmic curve or the exponential curve can never reach (touch) the vertical or the horizontal axes, respectively. It is called asymptotic in mathematics.
I had also webmaster problems caused by that huge-jackpot frenzy. Many more-than-usually visitors were referred to my Web site. Not to mention that I am the only author of the only software that truly handles Powerball, Mega Millions, Euromillions: MDIEditor And Lotto WE. The ZIP file had way too many simultaneous tries to download it. The result was: Complaints from registered members of my website. MDIEditor Lotto triggered an error (Temporary Unavailable). After big headaches, I realized that the zip file got corrupted! Among other bad things, the GoDaddy webhost even modified all my Web pages by adding the size to all images on every page! I recreated the package and uploaded it. MDIEditwe.zip can be downloaded now without problems.
I understood the pain and frustration of others, but some people were way too nasty. I ain't guilty if they don't win!