This is a follow-on to our recent article, mega millions changes A, which explored the different game change statistics
and the implications so far for increased gaming revenue. Apologies if there were any RSS hiccups given tech changes on my side. Now we happen to be
writing this as the current jackpot continues to rise slightly over $300m and
has the potential to the be first “large” jackpot since their April
modifications. Those changes included the
first ball count change in nearly 8 years, and the first price changes in
nearly a dozen years. As promised we wanted
to delve just a little deeper into the probability seeing the slowdown
discussed in the prior article.
More acutely one way to look at this slowdown is to ask the
chance of seeing such binary results such as 6 large jackpots in 2024 [between both
Mega Millions and Powerball], and now 1-2 large jackpots in the first half of
2025. A Powerball was won on New Year’s
2024, the bulk of tickets clearly bought in 2023. As a result we can consider 2024 to have 5.5
large jackpots as a statistical handicap.
Similarly we’ll consider 2025 so far to have 1.5 jackpots, since it’s
not completely fair to consider the Mega Millions’ revenue to all be generated
in the first half of 2025, when we don’t know the ultimate size and timing of
the current game’s win. This could be
big, or it could all end sub-$350m on Tuesday June 24.
Using 0.5 outcome values also changes the statistics slightly as we’ll discussed at times in this Statistical Ideas website.
Math parts
Now there is an expression known as the maximum likelihood
estimator that provides the best number of outcomes expected, given the number
of monthly trials. Since we are looking
for a single estimate at this time it is as easy to apply as a Bayesian
approach where we are trying to solve for the value across both years.
For 2024 clearly 5.5 large jackpots over 12 months, equals a
rate of 5.5/12 [or 0.46 large jackpots] per month. So now the cumulative binomial distribution for
seeing 1.5 or fewer 2025 jackpots so far is the sum of all outcomes, assuming
the 2024 rate:
p[X=k] = nCk * pk * [1-p]n-k
where n=5.5, and for all k of 0, 1, and part of 2
We’ve covered combinatorics quite a bit on
this site too, with some of the earliest articles shared widely on the New York Times, CNBC, and Bloomberg TV. Because we are using a
fractional n of 5.5, we will need the gamma [Γ]
function math described in the top link above.
But then we need to resolve the k of 1.5, with
approximation.
For one large jackpot [k=1] the probability of
seeing the 2025 “slowdown” using 2024 statistics is 24%. And for
two large jackpots [k=2] the probability of seeing the 2025 “slowdown”
using 2024 statistics is 57%. The answer
is closer to 24% than 57% simply because we can’t go higher than the 50% “expected”,
given we have observed a slowdown.
Conclusion
It is essentially a close call right now. There is a slowdown indeed in large jackpots
being fueled with underlying ticket sales, but it’s a draw whether such a
slowdown in revenue is a distinct feature of 2025 or something that can be
easily reversed in the many months ahead.
This is again where long-view copula statistics can be
helpful, especially in understanding the business dynamics governing the “sudden”
slowdown.
Additionally
This would normally be its own stand-alone article but in
the interest of time ahead of Tuesday’s New York City democratic primary
election it was important to put some context behind the many polls and betting
sites seeing to provide information on the election.
We used a monotonic-t, with autocorrelative stochastics to show most advanced approach for the City. Further, below we simulate 1000 so that you get a sense of who would when, and in which round. One can see there’s a decent 1 in 5 odds of a straight Andrew Cuomo win right at the jump, but otherwise it's a dodgy 2-man scuffle that can spill 8 or so rounds. In the latter case Zohran Mamdani would have a 1 in 3 or so odds of winning in these later rounds. For example, seen in this simulation here he would get about 70 round 9 wins out of 1000 simulations, with Cuomo getting just over 100 wins round 9 wins.
There was no scenario out of 1000 simulations where any of the other 8 major candidates would win in any round. And 5 simulations out of 1000 where we result in essentially a final round recount/tie! For more on other life's applications of copulas, please see my best-seller from last year [linked below].
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