I recently discussed several models for a Jack Whitaker beating jackpot and I think the last one is probably a better one than the first, which noted linearity up to the present.

If powerball rolls over again tonight, which is a better than even shot, it will be interesting to see if the rate of increase breaks from linearity. If so, "lotto fever" has clicked in. Because of the newly lengthened odds, the existence of "lotto fever" does not necessarily create a near certainty that the jackpot will be won. In fact, if the MM sold $200M in tickets (a cash value increase of $64M in a single drawing) there would still be a roughly 30% of the jackpot rolling again. This means that the occurrence of a winner would be probable, but far from certain.

The introduction of the cap in powerball will make my calculations far less accurate though since it will become more difficult to predict sales. If that occurs I will probably default to assuming that sales will be similar in each drawing until the jackpot occurs. All mathematical modeling relies on past experience, and the absence of such experience reduces accuracy. With the new odds, Powerball to some extent is in uncharted territory. Personally I think the cap is silly.

Because we have recent history on large jackpots in MM, that being the last run, my estimates are better here than they are there in this region of large jackpots. Hopefully though, they remain however useful starting approximations.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: October 12, 2005, 8:29 am - IP Logged

The Megamillions has rolled over again, and the cash value has risen from $38.6M to $46.0M, the annuity value from $65M to $77M. In this prize region on the last run, sales ran about $22M. The probability distribution for numbers of winner is as follows:

0

88.23%

1

11.05%

2

0.69%

3

0.03%

I would estimate that the probability of matching the $250M annuity jackpot that ended the last run would be between 22-25% and the long range probability of exceeding it would now be 14-15%.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: October 15, 2005, 1:21 pm - IP Logged

The Megamillions has rolled over again, and the cash value has risen from $46M to $53.3M, the annuity value from $77M to $90M. A reasonable estimate for sales expected is $24M. If this proves correct, the probability distribution for numbers of winner is as follows:

0

88.23%

1

11.05%

2

0.69%

3

0.03%

I would estimate that the probability of matching the $250M annuity jackpot that ended the last run would be between 25-30% and the long range probability of exceeding it would now be 15-18%.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: October 19, 2005, 8:41 am - IP Logged

While Powerball is getting all the attention, Megamillions is quietly growing.

The Megamillions has rolled over again, and the cash value has risen. Based on sales, the cash value reached from $54.6M and is expected to rise to $63.9M, the annuity value from $90M to $108M. This means that the lottery expects to sell about $29.3M tickets. From these values I estimate the probability of various numbers of winners as follows:

0

88.23%

1

11.05%

2

0.69%

3

0.03%

I would estimate that the probability of matching the $250M annuity jackpot that ended the last run would be just under 35% and the long range probability of exceeding it would now be near 21%.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: October 22, 2005, 11:49 pm - IP Logged

The Megamillions has rolled over again, and the cash value has risen. Based on sales, the cash value reached $64.9M and is expected to rise to $76.5M, the annuity value from $108M to $128M. This means that the lottery expects to sell about $29.3M tickets. From these values I estimate the probability of various numbers of winners as follows:

0

81.24%

1

16.88%

2

1.75%

3

0.12%

4

0.01%

I would estimate that the probability of matching the $250M annuity jackpot that ended the last run would be near 44% and the long range probability of exceeding it would now be near 26%.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: October 26, 2005, 8:26 am - IP Logged

The Megamillions has rolled over again, and the cash value has risen. Based on sales, the cash value reached $75.9 and is expected to rise to $87.2M, the annuity value from $128M to $147M. This means that the lottery expects to sell about $38.7M tickets. From these values I estimate the probability of various numbers of winners as follows:

0

80.23%

1

17.67%

2

1.95%

3

0.14%

4

0.01%

I would estimate that the probability of matching the $250M annuity jackpot that ended the last run would be in the neighborhood of 53% and the long range probability of exceeding it would now be near 32%.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: October 29, 2005, 9:35 am - IP Logged

The Megamillions has rolled over again, and the cash value has risen. Based on sales, the cash value reached $87.9 and is expected to rise to $85.9M, the annuity value from $147M to $165. This means that the lottery expects to sell about $35.1M tickets. From these values I estimate the probability of various numbers of winners as follows:

0

81.89%

1

16.36%

2

1.63%

3

0.11%

4

0.01%

This run is producing, overall, higher sales than the last run, which has complicated an understanding of what the odds of a run exceeding the last Megamillions run's $250M annuity jackpot will be. The overall sales difference is $15.9 million, and only the two most recent drawings have fallen short of the corresponding drawing in the previous $250M annuity run. I'd estimate that the likelihood of two more rollovers approaching but perhaps not matching exceeding the $250M annuity prize, but accomplishing it with one less drawing overall, is around 64%. The probability of 3 rollovers, I'd estimate at being around 35-40%. A third rollover would almost certainly exceed the $250M figure and approach $300M.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: October 31, 2005, 11:45 am - IP Logged

I think I want to refine my impression of the probabilities of reaching a new record jackpot, based on the following considerations.

Usually long runs are distinguished by a breakout rollover wherein sales, after proceeding in a linear fashion at a slow regular rate of increase, suddenly begin to grow exponentially until a prize is won.

Recently, on this run, sales have been performing somewhat erratically, for some reason, but still some semblance of linearity, albeit "noisy" linearity, has been maintained.

I note that in the previous run that breakout rollover was reached when the prize hit $177M annuity, whereupon the next two draws, another rollover and finally one on which the prize was won, showed exponential sales growth graphically. Although the prize is not quite as high now, and although the lottery is predicting linear sales by the size of the jackpot, I think we may see larger sales on this run than anticipated, somewhere in the mid 40's. This would bring the probability of a rollover down into the high 70's in percentage terms, but raise the actual current jackpot higher.

I think that two more rolls would leave us just shy of the $250M annuity jackpot. Playing a bit with the numbers, I'm estimating that the probability of two rolls is in the neighborhood of 50-55%. I'd estimate the probability of a third roll after that, blowing past the $250M annuity figure, at being, overall - including the previous rolls - at being somewhere in the neighborhood of 30-35%

This is my best guess. We'll see starting tomorrow.

Sparta, NJ United States Member #18331 July 9, 2005 1977 Posts Offline

Posted: October 31, 2005, 3:43 pm - IP Logged

Watch the local news. If they talk about the annuity number longer then 3-seconds, there will be a hit. If I had the time and the money, I would ask each player I saw if it was their first time. If they said yes, I would buy the same number. That would be playing the probabilities.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 1, 2005, 10:46 am - IP Logged

I have played with some numbers again, and modeled the current MM run using a linear extrapolation of the previous run to try to model what a very large run might look like starting from where we are now. The first column is the size of the advertised annuity jackpots we would see (obviously they're too precise) The second colum gives the odds that jackpot of that size would rollover again. The third column gives the probability that all the jackpots, including that one would rollover. Thus we have a 16% chance according to this model, of seeing a $382M jackpot, and a 7% chance of seeing a $471M jackpot.

$575,576,307.50

0.321984

0.87%

$471,808,242.96

0.376128

2.71%

$382,272,072.89

0.439376

7.20%

$306,967,797.28

0.51326

16.38%

$245,895,416.14

0.599568

31.92%

$199,054,929.45

0.700389

53.2%

$166,446,337.23

0.760034

76.00%

The formulas used to generate these numbers used a linear function to approximate an exponential function, which is not too bad over short ranges. The lowest datapoint represents the model output to show where we are now, and it's not too far off. It is worth noting that in the highest jackpot, $575M, the most probable number of winner would still be one winner.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 2, 2005, 9:28 am - IP Logged

The Megamillions has rolled over again, and the cash value has risen. Based on sales, the cash value reached $97.9 and is expected to rise to $113M, the annuity value from $165M to $192. This means that the lottery expects to sell about $47.4M tickets. From these values I estimate the probability of various numbers of winners as follows:

0

76.36%

1

20.60%

2

2.78%

3

0.25%

4

0.02%

Looking at my last post, we can see that I underestimated where the jackpot would be on this drawing by $7M annuity. However this is not totally unexpected since this is mathematical modeling of what will become an exponential function by a localized linear function. Actually, graphically, there is no evidence of lotto fever on the last draw. Although there were two outliers at drawings 12 and 13, a linear (no lotto fever) rise in jackpot values still is consistent.

I still expect that a rollover again is probable, as shown above, and that the next jackpot will be $245M +/- 10M. The linear approximation will over estimate at the beginning and under estimate at the end. It may become necessary, if a number of rollovers occur, to readjust the slope of the approximating line. For now I will adjust the spreadsheet used to generate those numbers, excluding the last rollover of course, which as gone from being a probable event to a certain event. I will also treat the coming jackpot as if it reached the modelled amount $199M, which it may do if a little more lotto fever kicks in (in which case the rollover probabiliy will drop from 76% to 70%.) We'll see. Here are the long range rollover probabilities.

$575,576,307.50

0.321984

1.15%

$471,808,242.96

0.376128

3.56%

$382,272,072.89

0.439376

9.47%

$306,967,797.28

0.51326

21.55%

$245,895,416.14

0.599568

41.99%

$199,054,929.45

0.700389

70.0%

Some comments: We see that we now have a 40% chance of breaking $300M. A jackpot at $306M would actually provide a higher cash value than Jack Whitaker's prize did, something that the recent "record" powerball did not do, because of the use of the cap, and the changed Powerball annuity structure, which inflates the annuity value. Specifically a $306M jackpot would have a cash value of over $185M. Jack's jackpot had a value of $171M. We have about a 22% of breaking the all time annuity jackpot record and reaching a jackpot close to $400M, and roughly a 10% chance of reaching a jackpot of close to $500M. The $306M modelled jackpot would be the first with even odds between there being a winner or winners and a rollover. At no level, according to this model, are any number of winners greater than one more probable than 1 or zero winners.

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 5, 2005, 2:03 pm - IP Logged

The Megamillions has rolled over again, and the cash value has risen. Based on sales, the cash value reached $113.1 and is expected to rise to $131.4, the annuity value from $165M to $192. This means that the lottery expects to sell about $57.9 tickets. From these values I estimate the probability of various numbers of winners as follows:

0

71.93%

1

23.70%

2

3.90%

3

0.43%

4

0.04%

The lottery's own last prediction of how many tickets they would sell was unusually close. Graphically, one sees some evidence of lotto fever, but it is not as spectacular as I anticipated in my last post, the graph resembling an exponential curve, but only the early portion where the derivative (the rate of change) is lower than it is as the function is applied to higher values. The actual sales for this drawing were closer to the $172 million jackpot of the last run, than it was to the $200M jackpot of that run.

I'm not sure of what the explanation is, probably jackpot saturation.

My predictions of long term behavior like those of the weatherman or weatherwoman, they are only as good as the model itself, and, as we have seen in the current hurricane season, the models can and do break down with differing intitial conditions, as is the case with global climate change and here, changing odds and new purchasing populations.

Still, I am still confident that, over the long term, if the jackpot is NOT won, an exponential function will nonetheless ultimately prevail, and so I will substitute the known data into a similar mathematic function as used to generate my last post, ie, a linear approximation to the local exponential region.

The new (somewhat more speculative than the data for the next drawing) long term modeled probabilities, for what they're worth, look like this:

$299,821,993.96

$491,329,924.99

0.376128

3.56%

$245,184,749.87

$401,793,754.92

0.439376

9.47%

$199,232,169.09

$326,489,479.31

0.51326

21.55%

$161,964,251.63

$265,417,098.17

0.599568

41.99%

$133,380,997.48

$229,089,499.49

0.700389

70.04%

Thus we have a 42% chance of seeing a jackpot over $300M and around a 10% chance of seeing a jackpot approaching $500M according to this model.

Here the first column is the size of the cash jackpot, the second the annuity value, the third, the rollover probability for a jackpot of that size, and the last column is the overall probability of getting past that level from where we are now.

BILOXI, MISSISSIPPI United States Member #19651 August 3, 2005 621 Posts Offline

Posted: November 5, 2005, 2:35 pm - IP Logged

i saw charts on this small roll overs are good for one person far 365 million the odds are at 20-30% i hope and the will go up with no winner tuesday whats the odds with 2 winners if a $100 million roll over happens you good thanks

New Jersey United States Member #21206 September 4, 2005 949 Posts Offline

Posted: November 5, 2005, 4:29 pm - IP Logged

Quote: Originally posted by JAG331 on November 5, 2005

Hasn't the jackpot already rolled to $225 million? So we have a 100% chance of reaching near a $229 million jackpot.

The number in the second to last column is the probability of the jackpot of that size rolling over, not the probability of reaching that jackpot. The last column is the odds of all the jackpots rolling over, including the one of the size listed.