Lotteries are typically Loser’s games; it is a sucker’s game. The odds and payoffs are statistically designed to favor the house. That has changed for the current PowerBall lottery because nobody has correctly scored the correct 6 numbers for several drawings, and the major jackpot has accumulated to record levels.
In simplest terms, not even including minor payoffs, the expected value of the prime prize exceeds the cost of participating by a respectable multiple. Of course, the odds of winning that prize are still minuscule.
Using the PowerBall rules, the odds are a tiny 1 to 292,201,338 to win the grand prize. Those odds are simply calculated based on the PowerBall rule set. Six balls are randomly drawn. The initial 5 balls have 69 possibilities and draw position doesn’t matter. The final ball comes from 26 possibilities and its sixth position does matter.
Therefore, the total number of possibilities is a combination of 69 possibilities taken 5 times. That combination must be multiplied by the last ball probability of 26 options. That’s (69 X 68 X 67 X 66 X 65) / (1 X 2 X 3 X4 X 5) all times 26 which equals the 292,201,338 winning odds. The calculation is not difficult; winning is another matter.
The jackpot is anticipated to be 1.4 billion dollars. Hence, the expected return is 1.4 billion divided by 292,201,338 or 4.49 dollars. Given that a single chance costs 2 dollars, the payoff ratio is atypical and attractive. This cost/benefit tradeoff comparison is positive which is unusual for lotteries.
I am not a gambler, but the current game challenges me to abstain. Opportunity seems to be knocking. As William A. Ward said: “Opportunities are like sunrises. Wait too long and you miss them”.
So, I am in the game for a few bucks despite the long odds. The odds of hitting a hole-in-one are much more favorable. Unfortunately, the likelihood of being hit by lightening are not rare compared to the lottery although in that instance specific location and State-by-State variations matter.
The Internet offers sites that have tabulated all the numbers ever selected on PowerBall for whatever that’s worth. If you believe that is worthwhile, you are falling victim to the Gambler’s Fallacy trap. Gambler’s Fallacy is believing that if something happens recently or more frequently than normal during some period, than it will occur less frequently in the future. That’s a regression-to-the-mean theory, which is not applicable to any independent dice throw or random ball drawing. If fairly executed, lotteries are purely random events.
I don’t subscribe to the Gambler’s Fallacy. As I stated earlier, I am now playing the Sucker’s game. In this instance, I see it as a rare opportunity.
Good luck to me and good luck to you if you decide to join our greedy mob. The PowerBall drawing is on Wednesday.
Without that risk, in theory, you could potentiall buy every possible combination for less than $600M and walk away with a guaranteed $1.4B.
There are some papers, I believe, which suggests buying into a large pool for smaller payout is more optimal than single buys. But all of them fail under the risk of multiple independent winners.
The "lottery thinking" is not uncommon in investing. There is some empirical evidence that people (even the smartest ones) are biased towards very large outcomes with much lower odds than smaller outcomes with better odds. If you do any kind of start up fund pitching in the Bay area, especially with angel investors, you get to experience (and exploit) it.
Or a way to get some of your welfare tax dollars back.
Or a way to keep the poor poor
Or as the price to dream.
I enjoyed your post, especially putting a fund startup spin to it. I also liked your global money balance demonstrating a positive reward if you covered the entire waterfront. Thank you.
I did consider the possibility of dual winners. Not being a probability professional I did a rudimentary calculation that I hope captures it's unlikely happening. I simply multiplied the probability of a single happening with it happening again. That product is astronomically small so I discounted it as not happening. I recognize that is a non-conservative assumption.
Once again, thank you for your insights.
Thank you for sharing your strategy approaches.
You and I are on the same page here. I too wanted to avoid number picking biases which conceivably generate multiple winners. So earlier today I allowed the computer to randomly select my entry numbers.
If I'm a big winner don't count on a dinner. Sorry about that!
This is worth a read. Here's an excerpt:
"In October, the consortium of states that runs Powerball approved a series of rule changed that made it much harder to win the jackpot. Under the new rules you select five of 69 numbers, up from five out of 59 numbers. The choices for the Powerball was actually reduced from 35 to 26. Still, this decreased the odds of winning the jackpot from 1 in 175 million to 1 in 292 million.
The purpose of this change was to increase the chances that there would be no grand prize winner for any given drawing. When this happens, the prize pool rolls over creating giant jackpots. At the time the rule changes were first floated in July, FiveThirtyEight estimated that the chances of a $1 billion prize pool increased from 8.5% to 63.4% over a given five year period."
Also, this: "And it’s those who can least afford to lose any money who are most likely to be buying tickets. Low-income people account for the majority of lottery sales, while sales are highest in the poorest areas. One study found that the poorest third of households buy more than half of the tickets sold in any given week."
Lottery is a concept. Donald trump won the lottery when he was born to his parents. I think he bought 100000 tickets. This should be a debate question.
Thank you all for participating in what turned out to be an informative and fun exchange. I had doubts about the interest level before posting.
Wow! Holy Cow Batman, the outcome was spectacularly unexpected: a trifecta of winning tickets, yet unclaimed. That highly unlikely event (probability nearly zero) is equivalent to being hit by both a lightening strike and a crashing airplane while simultaneously making a hole-in-one. That never happens in theory, but it did in practice. That’s the nature of rare, low probability events; they are memorable because of their unexpected occurrences.
Not surprisingly, I did not win the royal prize. I went bust. I was defeated by the lottery odds.
I had never played the lottery before this “opportunity”, and I do not plan to play again unless the expected return relative to cost becomes positive once again. According to those who know and follow this lottery’s rule changes, this is likely to happen more frequently now. In making my decision, I only considered the odds and payoffs, and did not ponder the ethics of the concept.
By playing, I was seeking entertainment with some excitement. The lottery delivered on both scores. I learned some important lessons from the game itself, from its history, and from you folks. You offered some useful, practical advice on how to tilt the odds just a little more towards the participant. It was all worth the price.
It’s amazing how much behavioral economics thinking impacts participant actions. For me, things like Daniel Kahneman’s Prospect Theory motivated me to buy a few tickets. The asymmetry in that Theory around its neutral point was precisely the finding that encouraged my chance taking. That finding suggests that the payoff prospects must exceed the risk penalties by about a factor of two before the wager is acceptable.
One of the tactics that you recommended was to allow the computer to randomly select numbers. That tactic eliminates the bias in choosing well-traveled patterns that produce multiple winners. I did that. Also, there is a little of Nassim Nicholas Taleb’s Black Swan logic embedded in some of the other tactics you offered. His strategy to accept small, slow bleeding losses while going for the low probability jugular big payoff is why many lottery regulars play. Taleb has recommended that same policy for some portfolios.
I sure don’t have the answers. One investment maxim is that the method, not any specific outcome, matters most. That’s obviously not true when playing any lottery. Outcome rules the day.
So ends my lottery experiment. Perhaps we’ll do this together again, then again, maybe not. Thanks again for your terrific inputs.
... of getting at least one number right in the winning combination. That is a powerful demonstration of the odds.
Not sure what you mean by "highly unlikely event (probability near zero)". Do you mean the event of three winners? If so, it isn't so. Higher the number of independent participants (same person won't repeat a number), higher the likelihood that there will be multiple winners.
In fact, if you have more people participating than the number of possible combinations the odds of two winners across all participants can be lower than the odds of any one particular number winning the lottery (the odds you calculated). Think about it. To illustrate, if two times the number of combinations bought the tickets covering all the combinations twice, the odds of any one number winning is still the same as you calculated, but the probability that there will be two winners regardless of the winning combination is 1 not even smaller as intuition might incorrectly calculate.
I don't know how many people participate in this mega event but if I were to base it on the amount of total sales, I suspect it is in millions and hence the likelihood of multiple winners can be higher than the odds of any particular individual winning. The only thing that reduces this likelihood is people bunching up around "popular" or "lucky" numbers. The higher the number of participants, less effect of that non-uniform distribution so max likelihood of multiple winners is reached asymptotically.
This is why no one is trying to game the system by choosing all combinations. There are significant number of people in the Bay Area capable of putting up the required amount for it in pairs or threes to double their money even with a small risk and the technology to get a "flash mob" to do the manual purchases. They would have lost money in this one, for example. The risk is actually high because the likelihood of multiple winners is non-trivial.
You're absolutely right that an increasing number of players shortens the odds significantly as that number grows massively. That was certainly the case for this lottery.
My example was poorly conceived. It was wrong for the point I was attempting to illustrate. As you correctly noted, the odds of multiple winners were much higher than I improperly calculated.
Thank you for your.fine insights.