# Is 538's math on the Powerball correct?

Here is the link to the page:

The author claims that there is a 91% of at least one winning ticket for tomorrow’s drawing. He assumes 428 million tickets sold based on the predicted jackpot of \$800 million. However, isn’t the main reason for the large jackpot that there were no winners over the past few weeks? If so, I would think that the 428 million ticket assumption is too high since many of those were purchased during previous weeks (and obviously not winners).

Also, for those doing the math at home, tickets are \$2 a pop, which would assume \$856 million of revenue if the 428 million ticket assumption was correct.

I don’t know about the number of tickets sold, but the odds of one of those tickets winning is easy enough to check.

The probability of any one ticket winning is 1 in 292 million. So the probability of that ticket NOT winning is 291,999,999/292,000,000. The probability of 428 million independent randomly selected tickets ALL not winning is (291,999,999/292,000,000)^428,000,000, which works out to 23% on my calculator. That means there’s a 77% chance that at least one ticket will win. Dunno how he got 91%, but he’s a stats guy and I’m not.

The jackpot is only a portion of the revenue. They’re looking at about \$300 million dollars added to the jackpot for tomorrow. So a third of the ticket revenue since the last drawing might be right. They have to give the money to the states to not spend on schools, give a portion to the places that sell the tickets, including bonuses for winning tickets sold, pay off the prizes below the top, take the operator’s profits, and have money left over to start the next drawing at \$40 million after I win this one.

In the article, he says that the odds were recently changed to 1 in 292 million from 1 in 175 million. If you run your same calculation using 175, you get an 8.7% chance of no winner, so that’s probably what he did.

What does that have to do with anything? If I have a ten sided die, there’s no law saying that I can’t roll a 1 ten times in a row. It’s not likely, but it can and eventually will happen. There’s no reason for today not to be that day.

Sorry, not following. The author is basing his assumption that there will be 428 million tickets sold on the fact that the jackpot is predicted to be \$800M. But the reason that it’s so high is because no one has won over the past few drawings. But the tickets for the prior drawings are moot since they can’t be used for tomorrow’s drawing. While they contribute to the high jackpot, they play no part in the probability that anyone will win tomorrow.

Ah, okay I misunderstood you.

I presume that his point was that the number of people has grown, but (since the length of the target lottery number hasn’t changed) the probability for any one ticket to strike hasn’t changed, so the probability of a win is higher.

The author is estimating that people will buy 428 million tickets.for the current drawing based on the size of the jackpot and how many sales there were for previous large jackpots. He isn’t calculating how many tickets were sold in the past to make the jackpot this size.

They made major edits to the original article when I posted the OP. I wonder if the original is available on the web.

Note: the cash value of the “\$800M” dollar jackpot is “\$496M”. With a 1:292M chance of winning and a \$2 entry cost, it still doesn’t theoretically pay even assuming you’re the sole winner (and the other forms of winning don’t matter). Not even at \$900M (\$558M).

And if you are a big fan of the lesser ways of winning, this isn’t the game for you.

What are you talking about? Do you mean it’s not worth buying 292 million tickets to win?

292 million tickets at \$2 would cost \$584 million. If you took the lump sum of \$558 million, you would have lost \$26 million!

Ah, but you would also win all possible combinations of four numbers plus the powerball, one five plus the powerball, etc. Every 26th ticket should have the winning powerball, and they would be worth \$4 each. That adds up to \$44,954,052. The lesser prizes would add to the win considerably.

Unfortunately, you have to pay cash otherwise I would totally do this. “I’ll take one ticket for every possible combination. Don’t cash this cheque until next week, please”.

16 cents on the dollar. Spend \$10,000,000 on powerball tickets and expect to get \$1,600,000 back, on average and before taxes, in prizes of \$1,000,000 or less.

How is the size of the jackpot computed? Some percentage of the previous pool plus some percentage of the new pool? How many tickets can be sold in the upcoming round before each ticket has an expected value less than \$2 ?

But I was referring to spending \$584,402,676 on 292,201,338 tickets with each one different (all the possible combinations). One would have to win all the possible combinations that way.

And especially now that the jackpot may be in the neighborhood of \$1.3B (~\$800B, I believe, if you take the lump sum), you’d come out well ahead if you won the entire jackpot.

Problem is, you’ve got a decent chance of having to split the payoff with one or more persons who also bought a ticket with the winning number. And if that happens, you lose nearly \$200M if you split the prize with one other person, and ~\$300M if you split with two other people.

What does this mean, btw? Just changing the number of balls? I seem to remember it always being six balls including the red one, though.

Did they change the number of numbers you can select from?

That’s what I assumed; I just used simpler numbers. Here are more detailed workings:

Total investment = \$584,402,676

Prizes won
With 5 white balls
Grand prize #1
\$1,000,000 #25 Total = \$25,000,000
With 4 white balls
\$50,000 #320 Total = \$16,000,000
\$100 #8000 Total = \$800,000
With 3 white balls
\$100 #20160 Total = \$2,016,000
\$7 #504000 Total = \$3,528,000
With 2 white balls
\$7 #416640 Total = \$2,916,480
With less than 2 white balls
\$4 #10801392 Total = \$43,205,568
Total winnings exclusive of Grand Prize = \$93,466,048 (or 15.9934% of your purchase)

Value of Grand Prize
Assuming 1 billion tickets are sold in addition to yours, and that all but yours are bought randomly
The probability you get the entire grand prize is 3.264%
The probability you split the grand prize 2 ways is 11.169%
The probability you split the grand prize 3 ways is 19.113%
The probability you split the grand prize 4 ways is 21.803%
The probability you split the grand prize 5 ways is 18.654%
The probability you split the grand prize 6 ways is 12.768%
The probability you split the grand prize 7 ways is 7.283%
The probability you split the grand prize 8 ways is 3.560%
The probability you split the grand prize 9 ways is 1.523%
The probability you split the grand prize 10 ways is 0.579%
The probability you split the grand prize 11 ways is 0.198%
The probability you split the grand prize 12 ways is 0.062%
The probability you split the grand prize 13 or more ways is 0.024%