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.