This week’s column is about the aftermath of the big Powerball drawing last week. Literally, the aftermath.
The drawing this past Wednesday took the jackpot all the way to $1.6 billion! There were three winners who will share that prize. We will take a look to see if my advice from last week not to even think about playing all 292 million combos (even if it were feasible) changed at all as the jackpot grew.
Before the drawing took place, I had the pleasure of reading numerous columns posted on the Internet about Powerball. One column, which was written several months ago, explained how the probability of a billion dollar prize was going to go way up with the changes the lottery made a few months ago to lower the probability of the jackpot hitting.
This analysis was well reasoned and went through a fair amount of detail to present the author’s case. Just a few months in, and we had the largest single jackpot to ever happen to Powerball.
Unfortunately, most of the rest of the columns made me want to just bang my head against the wall. There was one column about how you increase your odds if you don’t use quickpicks (based on the possibility the computer will pick the exact same set of numbers more than once).
Another talked about how your odds go up if you play in Pennsylvania (or was it Ohio?) because more winners have come from that state than any other! Are any of these writers mathematicians involved in the gaming industry?
One column I didn’t see suggested to players that as the jackpot climbed to over $1 billion they should buy as many tickets as possible as it was now in their favor. Of course, that led to another column (which I did see) to state this was foolish because each ticket has a negative expectation and buying more and more would only increase your anticipated loss.
At a $1 billion dollar first prize, this may no longer be true. Well, if there was a negative expectation, buying more won’t help. But the premise of the first column is that at $1 billion, there is now a positive expectation. The question is, how true is this and whether it paid to spend $585 million to buy every ticket to attempt to win the $700-$800 million jackpot?
At those levels, this became a very iffy proposition. The stated jackpot is an annuity. The cash prize is only about 60% of that. Of course, if you buy $585 million in tickets the jackpot will likely go up another $300 million or so (maybe more).
So, 60% of the new total would be about $650 million or so. You’d also win back about $100 million in smaller prizes (16% of the total spent). You’re profit is about $150 million. The picture all falls apart, however, if you have to split the jackpot with even so much as one other person.
But now, the jackpot was $1.6 billion. If you buy $595 million more, the jackpot will be almost $2 billion. Again, that is an annuity, so the real take home prize in cash would be $1.2 billion. Even if you have to split that with one other person, you’re still up about $150 million (don’t forget about the $100 million you win in “consolation” prizes).
So, going into it, how likely were you to have to split the jackpot at least one way? To know this we would need to know what percent of all the combinations had been purchased. According to one source, with a few hours to go, almost 86% of the combinations had been sold. This makes the calculation very easy. If you bought all the tickets at that moment in time, you’d have a 14% chance of winning the prize all by yourself.
But, we’ve already said sharing the jackpot with one other person still leaves you doing well. So, what were the odds you’d have to share it with more than one person? The formula here is a bit more complex and requires knowing how many total tickets were sold.
We know about 250 million combinations were covered. I believe I saw that more than 400 million tickets were sold with several hours to go. If this increased all the way to 500 million, then on average each sold combination had two tickets covering it.
This, of course, is just an average. It is well known lower numbers are picked more often (from birthdates and anniversaries), so it is hard to know the true likelihood. What we do know is in the end three people won, which means had someone purchased all the tickets, the jackpot would’ve been split four ways. This would’ve left this buyer with a total of about $400 million (cash) back on his $595 million dollar investment. It would not have been pretty.
At $1.6 billion, it is not clear whether each ticket had a positive or negative expectation. If I had all the information available to me, it would be easier to calculate for sure. But as is always the case, there is a reason why they call it gambling. Next week, I have one more Powerball related column that will cover the probability that my son’s 10 tickets performed as horrendously as they did!
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Elliot Frome is a second generation gaming analyst and author. His math credits include Ultimate Texas Hold’em, Mississippi Stud, House Money and many other games. His website is www.gambatria.com. Email: [email protected].