Risk and Reward: How Do You Choose?

NOTE: The following post discusses the New York and multistate lotteries as an exercise in probability and decision-making. It does not constitute financial advice or recommendations. You must be 18 years or older to purchase a lottery ticket. If you are struggling with a gambling addiction, please call the HOPEline 1-877-8-HOPENY (1-877-846-7369) or text HOPENY (467369)

I’m fascinated by how we think about risks and rewards in various contexts. The lottery provides a particularly compelling example for me. There are two major jackpot games in the US, Mega Millions and Powerball. As I write this, the Mega Millions jackpot drawing for tomorrow has a top prize of $600 Million, while the Powerball has a top prize of $550 million.*

Suppose you want to shortcut this whole school/work thing and just get to the living large part by buying a bunch of lottery tickets. You might as well go for the Mega Millions, since it has a 10% higher payout, right?

Not so fast, you say. I’m smart and took stats in high school, let me compare the odds of those jackpots. The Powerball odds are 1 in 292 Million, while the Mega Millions jackpot is 1 in 302.5 Million. Okay, that yields an expected value of $ 1.88 for the Powerball per $2 play, and only $1.98 for the Mega Millions. So our gap is down to around 5%, but still argues in favor of the Mega Millions.

Ah, but that’s not the only prize. Even if we don’t win the jackpot, we might at least recoup our costs with some smaller prizes (and maybe even come out ahead). Let’s compare those, first for the Powerball:

PrizeOddsExpected value
$ 1,000,00011,688,053$ 0.09
$ 50,000913,129$ 0.05
$ 10036,525$ 0.00
$ 10014,494$ 0.01
$ 7579$ 0.01
$ 7701$ 0.01
$ 491$ 0.04
$ 438$ 0.11
Total Non-Jackpot EV $ 0.33

And now for the Mega Millions:

PrizeOddsExpected value
$1,000,00012,607,306$ 0.08
$10,000931,001$ 0.01
$50038,792$ 0.01
$20014,547$ 0.01
$10606$ 0.02
$10693$ 0.01
$489$ 0.04
$237$ 0.05
Total Non-Jackpot EV $ 0.25

Okay, now we’re getting somewhere interesting! So the Mega Millions Expected Payout (with Jackpot) is $ 2.23, while the Powerball expected Payout (with Jackpot) is only $2.21. So even closer, but still argues for the Mega Millions. And, since it’s higher than $ 2, should you put all of your assets into lottery tickets?

Well, there are a few more considerations. First, the Mega Millions allows you to get two “Just the Jackpot” plays for $ 3.00, reducing your cost per jackpot play to $1.50. Since the expected “yield” on a ticket is the jackpot rate/your cost, or $1.98/$1.50, that bumps your rate of return to a whopping 32%, instead of the 11.5% from playing the full game ($2.23/$2.00). Statistically speaking, that’s a pretty resounding win. But there are two final wrinkles: first, you may have to split the jackpot with someone else. Even a 2-way split brings your expected value down below the threshold of indifference. In the absence of robust data on ticket sales, maybe you think you’re more likely to have to split the higher-value jackpot, and that pushes you to favor the Powerball option. Secondly, your odds of winning the jackpot are still only 1 in ~150 Million for a pair of tickets. Unless you’re spending a fortune, your odds remain very low. Your threshold of a “life-changing amount of money” may differ, but if $ 1 Million feels like it would change your life, you might have substantial buyer’s remorse if you match the first 5 numbers but not the “Mega Ball” (which is ~25 times as likely as that you match them all). In other words, even if you’re ready to go HAM and dump $300 into tickets for this next drawing, you’d only have around a 1 in 1.5 Million chance of winning the “Just the jackpot” prize, v. 1 in 80,000 of getting the $ 1 Million if you put that same amount into the traditional $2 tickets.

Continuing with that logic, though, if what you really care about is maximizing your chances of winning a “life-changing” amount of money, maybe your threshold is whatever would allow you to retire early and live comfortably, say $2 Million, or about what you’d earn over a 30-year career earning $65,000/year. Here in New York, scratch-off ticket reports show the total remaining prizes for each level, meaning that you can calculate the total number of winning tickets, and if one assumes a consistent ratio of losing tickets to winning tickets, the expected payoff.

In that case, your best option (if you can find it) is to buy $300 of game 1354, “$10,000/week for life” with a minimum jackpot of $10,000,000. There are ~137,000 winning tickets remaining across prize levels, and with your odds of a winning ticket at 1 in 3.23, that yields around 445,000 total tickets, of which one is still a jackpot winner. At $20/ticket, if you bought 15 tickets you’d have a 1 in 30,000 chance of winning $10M+, or nearly 2.5x your chances of winning a simple million in one of the drawing games. To make it even more attractive, there’s zero risk of having to split the prize, and the total expected value of a ticket is $33.28, or a 66% return on your expected investment, much better than the 32% from the “Just the Jackpot” so even the hard-core statistician should be happy. Of course, with < 3% of the original tickets remaining for sale, you may have a hard time finding a retailer who sells the game.

I’ve heard aspiring consultants make the argument for the equivalent of nearly every option in here, depending on how they think about seeking the highest expected-value payout, or the greatest chance of winning, and how they think about competitors’ reactions. Very few, however, will ask or consider the client’s appetite for risk independently of their own. If you expect to play a game or face a scenario a hundred times, you may be willing to take risks that a client who can’t afford to lose three times in a row shouldn’t. The probabilistic approach should be the start of a discussion about the client’s opportunities and underlying approach to their business, not the end of that discussion.

* These are the stated prizes; the cash values are less than those amounts, at $442 Million and $411 Million respectively, but for the purposes of this thought experiment I’ll use the stated values.  

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