Powerball Math for more than $1 billion

This Wednesday evening’s Powerball jackpot is predicted to top $1.3 billion. It got that big because Saturday’s jackpot was bigger than $900 million, and nobody won. Here’s a short video showing Saturday night’s Powerball drawing:

If you have really good vision, you would notice 69 white balls in the container on the left and 26 red balls in the container on the right. What are the odds of choosing all five white balls correctly and also choosing the red Powerball?

This is an important question because if you match all six on Wednesday night, you would win the jackpot — so this is literally a $1,300,000,000 question (probably more — the jackpot goes up more if more people play — $1.3 billion is the best guess right now).

The Powerball website says the odds of hitting the jackpot are a little more than 1 in 292 million:

power odds

That’s correct, but how does that math work? And how could we figure out the odds of other events happening?

This sort of math is called “statistics and probability.” The idea of probability is to figure out how likely it is that an event (or series of events) will happen. For example: “The chance of an average person living in the US being struck by lightning in a given year is estimated at 1 in 960,000” according to the Wikipedia article about Lightning strikes.

With Powerball odds of 1-in-292-million, you are more than 300 times more likely to be hit by lightning than you are to win the jackpot.

But it’s at least 300 times more fun to win at Powerball than it is to get hit by lightning… so let’s use math to understand the odds of choosing five balls out of 69, plus the red Powerball.

If the game involved picking just one white ball, we would have a 1-in-59 chance of winning. If they picked two balls, what would our odds be? Well, when there are two events, you multiply their likelihood of happening together.

To win the Powerball jackpot, we need to get all five white balls, plus the red Powerball. You might think that the odds of getting the five white balls is 1-in-69 raised to the 5th power. Let’s say I get the first ball right (woo hoo!) Once I do that, the odds of getting the second ball right are actually 1-in-68, because now there are only 68 balls left (I got the first one right, remember?)

So the odds of picking five balls out of 69 would be 69 x 68 x 67 x 66 x 65, right? Well, no — let’s say the winning numbers in Powerball are 10-20-30-40-50. The lottery does not care whether the numbers come out in the order of 10-20-30-40-50 or 50-40-30-20-10.  In fact, they could come out 10-30-20-40-50 and I would still win. 

So what I did in just multiplying 59 x 58 x 57 x 56 x 55 was to neglect to take into account the multiple ways that the winning numbers can come up. It turns out there are 120 different ways to select five numbers. Imagine thinking of all the different ways to order these letters:

Mathematicians have figured out that the number of ways to order “n” objects is something called n!, or n-factorial.

5! would be 5 x 4 x 3 x 2, or 120.

Factorial, probabilities, combinations and permutations are all explained extremely well in this great resource called Combinations and Permutations from the fine folks at “Math is Fun.”

For Powerball math purposes, the key formula from that resource is this one:

formula

With Powerball, we have 69 things (white balls) and we are choosing 5 of them. So “n” is 69 and “r” is 5.

So if we plug all this in, we get 11,238,513.

But that’s not 1-in-292 million. What gives?

That formula only calculated the odds of getting the five white balls — if you do that, you get a million dollars. To get the big money, you also have to get the red Powerball — so you have to multiply 11 million by 26, which works out to 292 million.

So now here’s an application question that matters right now: is Powerball a good bet with an expected jackpot of $1.3 billion?

Generally speaking, it costs $2 to get a ticket, and each ticket has a 1-in-292 million chance of winning.  So is that a good bet?  Well, it depends on what the jackpot is… a ticket is worth the investment only when the expected monetary value (EMV) of the ticket exceeds the purchase price of the ticket — in this case $2.

A $1.3 billion payoff means that your ticket is worth 1/292,000,000 (the fractional chance you will win) times $1,300,000,000, which seems to work out to about $4.50.

So right now it seems like a ticket is technically worth $4.50 (really a little more, since you can also win smaller prizes, but the math is complicated enough to just think about the jackpot).

But that’s wrong, because you don’t get the full $1,300,000,000 — you get a portion of that if you take a lump-sum payment… and then you have to pay some hefty taxes on your winnings. Time Magazine has a nice piece called The One Time It’s Mathematically Advantageous to Play Powerball that explains this in more detail.

When the take-home value of the jackpot hits about $584,000,000, the EMV is $2.  So basically, when the take-home value goes over $600 million, it actually makes sense — mathematically speaking — to buy a ticket.

Extending this logic, it seems like a great idea to buy up all the tickets. Sure, at $2 a ticket, it would cost you just under $600 million to do — but you would be guaranteed a $1.3 billion payout. Let’s go!!!

Hold on — here’s where math and reality diverge 🙂

Even if you bought a ticket a second — in reality, it takes much longer than that, and you would need breaks — it would take you 292 million seconds to buy up all the tickets and ensure that you get the winning ticket.  That works out to more than nine years.

And if someone else used the same strategy (or got really lucky), you would have to split the jackpot.

The LA Times has a good simulator that shows how it’s incredibly unlikely that you will win any money playing Powerball. But when the jackpot gets this high, it’s still fun to play — and to think about the math in Powerball.

And there’s ethics, too — consider the morality of state-sponsored lotteries. Who are more likely to buy lottery tickets? Rich people or poor people? A book written by a pair of Duke professors, Selling Hope, calls lotteries an “insecurity tax” and argues that such lotteries are regressive, as opposed to progressive.

But lotteries fund all sorts of worthwhile projects.

According to the North Carolina Education Lottery website, lottery funds have contributed $4 billion to education programs since it started. Should North Carolina fund its education programs partially through lottery revenues?

So many questions and so little time… I’m going to buy a few Powerball tickets 🙂

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About Steve Goldberg

I teach students at Research Triangle High School (RTHS) about US History. RTHS is a public charter school in Durham, NC, whose mission is to incubate, prove and scale innovative models of teaching and learning. The blog posts here reflect my own personal views and not those of my employer.
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2 Responses to Powerball Math for more than $1 billion

  1. sarahsmilesalot says:

    Hahaha! You need to go outside and play with your son, Steve. This is a great project for kids and I probably would have liked math a lot if I had teachers who thought and taught like you! But I didn’t and, therefore, you lost me at “If you have really good vision…” Hope all is well with you and yours. Living in the Asheville area is wonderful. Working a lot for Bernie and enjoy wonderful people, despite our horrible government. Happy New Year! Sarah (Ritter) Dooley

  2. sarahsmilesalot says:

    I meant to say our horrible NC government.

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