Luck, statistics, and probabilities go together hand-in-hand. Clint Eastwood, playing Dirty Harry, famously asked a bad guy who was about to reach for his rifle whether he felt lucky. I’m quite sure that the crook carefully pondered the nature of luck, probabilities, and expected outcomes before deciding not to grab his rifle!
A while ago, I did something shocking . . . something that I hadn’t done for several decades. Just like the thief in the Dirty Harry movie, I started thinking about luck. Yes, you guessed it: I bought a lottery ticket for the record-breaking Mega Millions Jackpot. This purchase is shocking for someone like me who knows statistics and is fully aware of how unlikely it is to win. Did I feel lucky? Or was I just a punk?
What Is Luck?
Luck involves probabilities and expected values for outcomes. The lottery is a good way to illustrate this. Based on the probabilities involved, the expected value of the return on your money is less than the money you spend on lottery tickets. In other words, you need to be lucky to beat the long odds in order to receive more money than you spend on lottery tickets.
The odds of winning that record $540 million Mega Millions Jackpot was one in 176 million. Those odds are often compared to the odds of being struck by lightning, which are one in 1 million for a given year, according to the National Weather Service. You were 176 times more likely to be hit by lightning than you were to win the Mega Millions Jackpot. If you bought 176 lottery tickets, and didn’t repeat a set of numbers, your odds of winning are roughly on par with being hit by lightning within a year.
I can vouch firsthand that you are more likely to be struck by lightning than win the jackpot. I’ve been hit by lightning, but sadly I have not won the lottery.
Hmm. If I get hit by lightning 175 more times, do I get some sort of cosmic credit to win the jackpot?
Good Luck versus Bad Luck
While being hit by lightning might be considered unlucky, I actually consider myself quite lucky because I was unscathed. While only 10% of those who are hit are killed, most of the survivors have persisting injuries. I had no problems except for a temporarily sore hand. A sore hand, you ask? That’s thanks to the umbrella I was holding at the time . . . more on that later.
Strangely, winning the lottery is often compared to all sorts of awful things: being killed in a car accident or plane crash, murdered, dying from flesh eating bacteria, and the like. It’s interesting that we compare something that we’d really like to things that we’d like to avoid. So instead, let’s compare two approaches to becoming a millionaire.
We’ll compare a lottery approach and a savings approach. For both scenarios, we’ll assume that you spend/invest $1,000 a month for 30 years, for a total expenditure of $360,000.
This web site used the probabilities of winning to estimate that you can expect to get back $0.50 for every dollar you spend on Mega Millions tickets, including the lesser prizes. If you spend the $360,000 on Mega Millions, you’d expect a return of only $180,000.
If you invest $1,000 per month for 30 years and you have an annual rate of return of 6%, you’ll end up with $1 million. Of course, systematic investing does not guarantee a million dollars, but the odds are far more likely than those of winning a million in the lottery.
I picked a 6% return because the worst-ever 35-year return per year for the stock market was 6.1% (1906-1941). The average 35-year return per year is 9.7%. If anything, I’m being conservative. I could not find 30-year rolling returns. However, even if you assume a 0% return, you’d still end up with twice the expected return of the lottery approach!
In a nutshell, based on reasonable expectations, you need less luck to become a millionaire through investing than through the lottery.
Good luck happens when you beat the odds and achieve the unexpected. If you beat the long odds and win the lottery jackpot, you are lucky. When lightning struck me, I beat the odds by not being injured.
Related post: Probability Fundamentals
Statisticians and Luck
Luck is generally not something that statisticians are comfortable with. We understand all too well that luck entails relying on results that are unexpected and rare. And we have a very clear idea just how rare they are! That said, we do include our own special notion of luck in our analyses. The closest statistical equivalent to luck that I can think of is how we handle error. Error is the difference between the expected value and an actual observed value.
For the lottery example, the expected value of the return for spending $360,000 on lottery tickets is $180,000. However, very few people will fall exactly at $180,000. The actual winnings center on $180,000 and spread around it. Most winnings will be close to the center but a few will be far in either direction. The distance between the expected value and the value that you personally achieve is your luck, or error. If you have very good luck, you’ll be far to the right of the distribution’s center with a positive error. Unlucky? You’ll be to the left with a negative error.
Related post: Understanding Probability Distributions
Statisticians Minimize the Role of Luck
Statisticians try to control, quantify, and minimize the role of luck (error) in our analyses. To accomplish this, we follow strict procedures to help ensure that our errors are symmetric and random. In our models, statisticians include appropriate variables that will account for the observed variance and reduce the amount of unexplained error. We do this so that we can obtain a clearer view of the true role that the variables play. We don’t want large errors mucking up the picture!
This is true in quality management as well. A quality analyst would never say, “We got lucky this week and had 10 parts fall within the spec limits!” Instead, they carefully measure, analyze, and control the factors that affect their product. The expected values of the process output are carefully designed to fall within the spec limits, and errors are kept to a minimum. In other words, analysts minimize the need for luck to almost nothing.
Related posts: How Probability Theory Can Help You Find More Four-Leaf Clovers and The Monty Hall Problem: A Statistical Illusion and The Birthday Problem.
Don’t Rely on Luck!
I generally prefer to not rely on luck. Instead, I think it’s better to identify and understand the factors that affect the expected outcome for any given situation. You then choose the scenarios that maximize the expected outcome. With this approach, you are working with the odds, not against them.
With that in mind, I no longer use umbrellas in thunderstorms (hey, I was young!). And my retirement plan does not rely on systematically purchasing lottery tickets. Conversely, it also doesn’t rely on beating the market, which involves too much luck.
Finally, if you’re wondering what might happen if a statistician were to systematically play the lottery, read here to see how one statistician successfully cracked the scratch lottery code. Like I said, statisticians want to minimize the need for luck.
Never underestimate the power of statistics!
Thank you so much for your answer. Indeed, I could not find a statistical reference that would define luck as the difference between expected outcome and actual outcome. Instead, I found a philosophy article that uses this definition to explain luck. I mention it here in case it is useful to someone else.
McKinnon, Rachel. “Getting luck properly under control.” Metaphilosophy 44.4 (2013): 496-511.
Thank you for your reply!
Thank you for the interesting post. You write “The distance between the expected value and the value that you personally achieve is your luck”. I am writing an article that operationalizes luck in this way. A reference would be useful to me. Do you know of any statistics book that explains luck in the same way as you do here? Thank you very much.
Jim Frost says
I don’t think you’ll find it described that way anywhere else. Certainly not in a statistics article/textbook. However, random error is the difference between the expected value and the observed value when you have a good model. Theoretically, the expected value incorporates the portion the variability that the model explains while the random error is unexplained. Why do some observations have outcomes that are better or worse than the expected outcome? You can call that “random error” but we think about it as luck! Very similar concepts. For lottery tickets, the expected outcome is that each ticket will lose money. However, there are invariably some tickets that win money. Luck of the draw!
Kathleen Cameron McCulloch says
I love your blogs. Very intuitive. I often send my students to your page!
Hi Jim! I’m obsessed with your blog. Not only does statistics interest me greatly, but your own personal touch on your posts is inspiring. Please keep up the good work jim, your blog is the highlight of my day 🙂
Jim Frost says
Thanks so much for your kind note! It means a lot to me. You’ve made my day!! 🙂
New blog posts will be resuming soon!
Saumya Srivastava says
That’s an interesting post, Jim. I remember a FRIENDS episode where they all buy lottery tickets and Ross draws the same lightning bolt analogy.
Jim Frost says
Thanks! That seems to be a very common analogy to use!