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"Do You Feel Lucky, Punk?"

Clint Eastwood, playing Dirty Harry, asked this famous question while confronting a bad guy who was about to reach for his rifle. I’m quite sure that the bad guy carefully pondered the nature of luck, probabilities, and expected outcomes before deciding not to grab his rifle!

A month ago, I did something shocking . . . something that I hadn’t done for several decades. Just like the bad guy 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 getting hit 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 hit 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?

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 2 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.

Lottery Approach

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.

Investment Approach

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, accumulating a million by investing is not guaranteed, 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 (which seems warranted, given the current economy). 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. I beat the odds when I was hit by lightning by not being injured.

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 should be centered on $180,000 and distributed 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.

Statisticians try to control, quantify, and minimize the role of luck (error) in our analyses. We follow strict procedures to help ensure that our errors are symmetric, random errors. We 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, the need for luck is minimized to almost nothing.

Closing thoughts

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!

 

Comments

Name: Wesigye Alex • Wednesday, August 13, 2014

Thank you so much.
I am doing a research on wage differentials, but when a run a dummy variable regression in Eviews. i found out that there was heteroscedasticity.
I would like to know to formulate weights for variables in Eviews and how do i identify the variable with that problem
thank you


Name: Jim Frost • Tuesday, August 19, 2014

Hi Wesigye,

It's hard to give general advice about a very specific problem. You really have to know your data to be able identify the source of the problem. In general, look for a variable that has a large disparity between the largest and smallest values.

Also, keep in mind that weights only correct the form of heteroscedasticity that is caused by a variable with a very large range. Another form is caused by an incorrect model, such as a missing predictor. Weighted regression won't fix that type of heteroscedasticity.

I run through a discussion of using weights and example of how to perform weighted regression here:
http://blog.minitab.com/blog/adventures-in-statistics/curing-heteroscedasticity-with-weighted-regression-in-minitab-statistical-software

I hope this helps!
Jim


Name: Wesigye Alex • Friday, August 22, 2014

Thank you Jim,
yes, i changed the model into a log-linear functional form and there was no more heteroscedasticity.

I would also like to know how to change the significance level say from 5% to 10% in Eviews.

Thank you!


Name: Jim Frost • Friday, August 22, 2014

Hi Wesigye,

I'm glad that you found the solution! As you've found, there are multiple causes for heteroscedasticity. Sometimes you have to change model, and other times you should use weights.

Regarding changing the significance level, I can't provide support for Eviews. However, in Minitab statisical software you can change the confidence level (which equals 1 - significance level) in the Options subdialog.

Jim


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