Mega Millions Just The Jackpot

Reply #19

cottoneyedjoe — Tue, 08 Aug 2023 02:19:03 GMT

They are getting smarter then, LOL. Supposedly they also learn from all the times people correct them when they get wrong answers. Maybe Chat GPT wised up since I last tried it.

A slightly harder trick question for the AI chatbots is What is the smallest prime factor of 2^333333333333333 + 1? The answer is 3, because 2 raised to any odd positive integer power plus 1 is a multiple of 3.

Reply #18

Murgatroyd — Tue, 08 Aug 2023 01:23:49 GMT

Not bad, but overly specific in its reason. An odd number raised to any power, not just an odd one, will result in an odd number.

Reply #17

Orange71 — Tue, 08 Aug 2023 00:30:13 GMT

cottoneyedjoe, very interesting indeed!

FYI, Microsoft Bing AI chat (an extension of Edge browser) got it right, as follows:

The smallest prime factor of 3^33333333333 + 1 is 2. This is because any odd number raised to an odd power will result in an odd number, and adding 1 to an odd number will always result in an even number. Since 2 is the smallest prime number and the only even prime number, it is the smallest prime factor of 3^33333333333 + 1. Is there anything else you would like t... [ More ]

Reply #16

cottoneyedjoe — Tue, 08 Aug 2023 00:06:05 GMT

All the popular publicly available AIs are very bad at math and explaining math. A few months back I tested Chat GPT with this trick question:

What is the smallest prime factor of 3^33333333333 + 1?

It gave some bull sh1t answer about not being able to factor numbers that large, but it missed the obvious: 3^N + 1 is always even for any non-negative integer N, so the smallest prime factor is 2.

Reply #15

Orange71 — Sun, 06 Aug 2023 18:21:14 GMT

Apparently I won't be using Bard anytime soon based on that answer. Frightening that many people will likely just believe whatever AI spouts out. The AI generator says it, so it must be true.

Reply #14

Murgatroyd — Sat, 05 Aug 2023 17:46:38 GMT

Wow, that was a load of nonsense. Never ask a text generator to do a mathematician's job.

Reply #13

Wavepack — Sat, 05 Aug 2023 00:10:25 GMT

Bard is a rebard.

Reply #12

mrtckw — Thu, 03 Aug 2023 10:59:33 GMT

Answered by Google. Bard

The expected value of a lottery ticket is the average of the possible prizes, weighted by their probability. For a regular Mega Millions ticket, the expected value is about $0.63. This means that, on average, for every $2 you spend on a ticket, you will expect to win about $0.63.

The expected value of a JTJ ticket is about $1.26. This means that, on average, for every $3 you spend on a ticket, you will expect to win about $1.26.

So, the JTJ ticket offers a... [ More ]

Reply #11

JustMaybe — Tue, 01 Aug 2023 20:14:01 GMT

Should we just go back to farming and let our hopes be dashed.

This is a pretty tall order

Reply #10

Orange71 — Fri, 21 Jul 2023 00:18:49 GMT

Ponder that the winning numbers for PB and MM are selected by mechanical lotto machines, not PRNGs. The PRNG algorithms only select Quick Picks. Any slight bias in the algorithm is going to have such a very minor effect on the Binomial assumption it won't practically matter. The players with unique combinations selected on play slips will have a more significant effect on deviation from the Binomial Distribution.

There was one Jackpot winner in last night's $1 Billion Power Ball drawing. Per... [ More ]

Reply #9

Wavepack — Thu, 20 Jul 2023 01:32:28 GMT

PRNG through obscurity has repeatedly been shown in the past to be a poor methodology. Someone here posted about exploiting weak PRNGs used in past lottery draws, to the point of making it profitable on average to play those PRNG weak lotteries.

The knowledgable should call out the lottos for not openly vetting their PRNG.

Open source software is now quite strong now. The PRNGs used in open software have been vetted for the most part (everything but the seeding

Reply #8

Orange71 — Thu, 20 Jul 2023 00:39:21 GMT

I really doubt that can be proved as far as the PRNGs lotteries use. You would have to have a database of numbers picked for all tickets sold and know which ones were Quick Picks, and then run the significance testing. You're not going to get that information. Also, the lottery administrators have no real motivation to make their algorithm open source, so I doubt you'll see that either.

Reply #7

Wavepack — Thu, 20 Jul 2023 00:29:06 GMT

There are numerous famous cases of PRNGs not being properly seeded, or the PRNG being weak. Any cryptographic algorithm based on those weak PRNGs can then be cracked in reasonable time. Windows had a weak PRNG for a long while.

In the lotto world, a bad PRNG or PRNG seeding would result in excess repeats and pick biases.

Reply #6

Orange71 — Thu, 20 Jul 2023 00:20:43 GMT

I make the assumption that every single play line is perfectly randomly selected and independent of all other play lines. I wouldn't get hung up on the PRNG algorithm used. Yes, no code is perfectly random, but any deviation here is likely to be miniscule and inconsequential.

Reply #5

Wavepack — Thu, 20 Jul 2023 00:07:31 GMT

Interesting that the EVJ peaks.

What assumptions did you make on repeat ticket picks?

The PRNG (pseudo-random number generator) used in the Lotto ticket generating machines have not been been openly published, which is concerning. If those PRNGs are not seeded with enough bits of entropy (randomness), or the PRNG has a weakness, then the PRNG will not have a uniform pick probability, resulting in more repeats than expected from the quick picks.

Reply #4

Orange71 — Tue, 18 Jul 2023 00:17:14 GMT

To your point about variance, yes, JTJ is obviously going to be higher than a regular ticket. Both are going to be a huge variance, regardless. So, if variance is the concern, don't play the games to begin with. You will get a few bucks back by playing a regular ticket and hitting the minor prizes, but as percentage of $ spent, it's still very low. The return is basically predicated on hitting the jackpot, which the average person more than likely would not hit in a thousand lifetimes.

Ther... [ More ]

Reply #3

Wavepack — Mon, 17 Jul 2023 18:28:28 GMT

I should of mentioned, for the variance question, that an unbiased mean estimator is used.

I ask the variance question because the answer makes explicit the ridiculously large sample size needed to get a reasonably low variance of the estimator of the Mega Millions EV. The normal lotto player has a relatively small sample size, so the applicability of this EV analysis is of limited value.

Reply #2

Orange71 — Mon, 17 Jul 2023 14:50:27 GMT

Those are good questions, and it's interesting to explore the mathematics behind those questions. For your first question, I did not assume one winner. That's why I used the term EVJ - Expected Value of the Jackpot (with EV being the statistical weighted mean essentially). That depends on (a) the Jackpot advertised value (I'm using cash value for my analysis), (b) the number of tickets sold, and (c) the number of tickets per player with unique combinations (i.e. tickets per player bought not usi... [ More ]

Reply #1

Wavepack — Mon, 17 Jul 2023 00:52:32 GMT

For those answers, did you assume just one jackpot winner? If not, what number of tickets did you assume were sold?

I did the 3rd calculation a while back, factoring in taxes. I remember that it didn't make sense, from an expected return viewpoint, to play Mega Millions until the jackpot was roughly $600M, so your results are in the right ballpark, and are most likely correct.

The other calculation of relevance, along these lines, is the number of plays needed for the variance of the EV... [ More ]

Mega Millions Just The Jackpot

Orange71 — Sun, 16 Jul 2023 16:06:31 GMT

In my state the lottery commission offers an alternative to the regular $2 Mega Millions ticket. It's called Just The Jackpot . (I'll call it JTJ for short.) You get two tickets for $3, but you can only win the jackpot. Lower tier prizes are eliminated. Two interesting questions arise: (1) At what Expected Value of the Jackpot does the JTJ offer a better expected value of return per unit bet compared to a $2 regular ticket, and (2) At what Expected Values of the Jackpot do a regular $2 ticket an... [ More ]