Feb 16, 2008

Not Having the Time of Your Life. And You’ve Never Felt This Way Before. I Swear.

A periodic portion of my real life job (or my “adult job” as friends like to call it) involves traveling to Universities in the Northeast to give job interviews. These thirty minute trials of courage are a freeform mix of hard and soft, allowing sage questioners like me to say and ask practically anything. Don’t get the wrong idea; I take my white collar decider-of-fates role quite seriously. Not only because I want to admire the people I see at the water cooler but also since the ultimate successes and failures of my chosen hires are a direct reflection on my own tastes and values.

Given that one important prerequisite of my job is a penchant for all things analytical it’s common during the interview to pose a question whose answer requires some combination of math, spatial reasoning, and outright common sense. This is the part students fear most. Our hypercompetitive society buries kids $250k into debt to have a shot at this singular not-until-you-answer-this-riddle moment. Imagine The Showcase Showdown, Final Jeopardy, and Double Dare Obstacle Course all rolled into one. Except what’s on the line here isn’t a tricked-out RV, trip to Paris, or cash windfall; it’s what you believe to be your future. And it’s not that we don’t care about all the other things they’ve accomplished but if two other students who are also trilingual, kite surfing national champions, Fulbright scholars, and graduating three years early get this question right, who do you think we’ll be logically obliged to choose?

“How many degrees separate the hour and minute hand on an analog clock reading 3:15?”
“If you painted the surface of a cube made up by 1000 smaller cubes, how many of the smaller cubes would have paint on them?”
“Which investment is most attractive: one that doubles in two years, triples in three years, or quadruples in four years?”
“If there’s an equal chance of rain or sunshine, what are the odds of three consecutive days of rain over the course of five days?”

Depending on how your mind works these questions might sound really difficult, really simple, or just really silly. Exercises like these are one of the prospecting tools used by financial companies, consulting firms, and political think tanks to sieve the not-so apparent analytical dynamos from the sea of “fools gold” straight-A bookworms.

The motivation for this note was to share a question of this sort that’s fascinated me since the day I heard it. “What are the odds of a once in a lifetime event happening once in your life?” First, I must express my condolences to the students who’ve been asked this in an interview. Questions that are infinitely more simplistic and discrete consistently confound the smartest of students. This one is tough. To technically answer this problem you’d have to read up on Simeon-Denis Poisson, the 18th century French mathematician whose work focused on the modeling of improbable events. I didn’t quite get that far in my studies (or was sick that day) and have chosen never to seek out how one would answer such an interesting theoretical query.

Romantic notions of fate and destiny are routinely suppressed by the clockwork nature of my analytical psyche. A psyche that rarely yields at the opportunity to expose unpopular truths or debase myth, superstition, and hindsight bias. This mental framework has blessed/cursed me to see the world as a cold, chaotic set of fluttering stereo equalizer-like probability distributions where strange coincidences are simply tail improbabilities bound to occur during the course of our lives. The “once in a lifetime” conundrum sheds a rare and strange light on the seesaw that balances my conception of hope vs. uncertainty. I’m thankful that my unrelenting analytical pitchfork is willing to leave this notion answer-less and wonder-full.


Patrick said...


1. 7.5 degrees
2. 488
3. Quadruples in 4
4. 1/4

Modestmerlin said...

P -

To maintain the sanctity of this post I will neither confirm nor deny the correctness of your answers.

bean said...

if you call it living....

1. 0 degrees (arent they on the same plane in that case?)
2. 480
3. doubles in two.
4. who knows

Erik said...

1. 2*pi/48 radians
2. 1000-9^3
3. triples in three (doubles in two and quadruples in four are the same rate or return)
4. 1/4

The probability a once in a lifetime event happens to you in a lifetime is 1-1/e. (that was difficult)

My favorite, which isn't as difficult: Say you have a bag with 1000 pennies. One coin has two heads; the others are fair. You draw a coin out of the bag and flip it 10 times getting 10 straight heads. What is the probability that the coin has two heads?

Erik said...

I just thought of a better one: what is the probability the sun will rise tomorrow?

Modestmerlin said...

E -

1. It looks like those degrees from MIT and Stanford paid off.
2. You're overqualified.
3. Check your answer #1 again.

Erik said...

1. Causality likely runs the other way
2. Actually no
3. Convert the units yourself, smartypants (now you understand #2)

My answer to 2 was wrong, too much shooting from the hip. I'm going to need your answer for the sun coming up: I can't figure it out.

Modestmerlin said...

E -

Yes, I didn't even check #2. Patrick's 488 is correct (there you go Bean). Your responses were too complicated and witty for me to even understand. As far as the sun goes there are a few Big Bang era assumptions I'm still lacking in order to solve the problem. I'll come back to you after I speak to my college astrophysics professor.

Erik said...

I got thrown out of my only banking interview for being a prick. That, and calling a trading floor a white collar sweatshop.

You've inspired me to write something on these types of problems. Won't get to it for a while, but I'll send it to you when I am done.

Modestmerlin said...

E -

White collar sweatshops?! A very interesting sentiment indeed. If you end up making t-shirts please send me one. And DEFINITELY send me your essay.

Anonymous said...

90 degrees
quad in 4
2.5% chance

Michael said...

You want the simple solution to your once in a life-time problem? Think of a flood.

Modestmerlin said...

T -

Why must The Technician speak in such terse terms? I guess that's just his way.

Michael said...

Just seeing if you're paying attention.

I think if you take life to be a collection of a near infinite number of events (which, in a way, it is) then the 1-1/e solution is true. However, many events that can be considered "once in a lifetime" can't, by their nature, be computed this way.

If the once-in-a-lifetime (OIL) event were like a slot machine jackpot, and your life is reduced to millions or billions of events that are like pulls on the great lever, then you can use that equation. On the other hand, some events are of such great significance and are so dependent upon pre-requisite events, that they don't have millions of trials.

Meeting the love of your life might be a OIL event, and it might be expected by 1-1/e, because you cross paths with millions of people and only meet The One, once. On the other hand, how many times can you see a total solar eclipse? Even with the best of resources and travel means, you're only going to see a handful in a lifetime, because that's all there are. So, for the average Joe, is this not somewhat of a OIL event? But it is not because it occurs once in a billion trials, but because it happens so rarely. And it CAN'T happen any more frequently. That's the key, since a jackpot COULD occur on two successive pulls on a slot machine.

How many times can you get left at the altar? How many times can you see your favorite, life-altering film for the first time? How many times can you watch your child walk for the first time?

No, these events are not rare because of low probability, they are rare because the confluence of events that come together to make them possible are so PRECIOUS.

So what IS the answer? I think it comes down to actuarial figures. The insurance companies figure these things out all the time. I thought immediately of the "50-year" flood. Defined as: a flood with a peak level so high as to statistically only occur once in every 50 years. There are 50-year floods, 100-year floods, 5-year floods, etc. (Crucial note: that doesn't mean a river has millions of floods a year, and the 50-year variety comes every 50 million trials.)

So if you OIL event is like a flood, rare and unique to a human life, then let's put it in those terms: it's the 78-year flood (let's just call 78 years your life-expectancy).

So, by this definition, every year has a 1/78 of such an event occurring. So the chances of it happening in your lifetime are: 1-(77/78)^78.

Or, approximately: .634491439

Wait, is that the simple answer, or the complicated one?

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