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Driving Quicker Takes Longer – Algorithm Soup

Driving Quicker Takes Longer – Algorithm Soup

2024-01-27 09:21:42

I usually drive between Boston and New Haven. Whereas on the street, I discover myself pondering a easy query: If my solely purpose is to reach as quick as doable, how briskly ought to I drive?

Ignoring issues like ethics (or fuel efficiency), the answer would appear to be easy. Drive as quick as doable. However there’s a catch: If I crash and die, then the subsequent 50 years (which I had deliberate on spending alive) are spent lifeless. It’s solely honest to rely this as a penalty in direction of the size of the journey.

So, the anticipated journey size isn’t simply {frac{text{distance}}{text{speed}}}. It’s really

{frac{text{distance}}{text{speed}} + Pr[text{Death}] cdot 50 text{ years}.}

How a lot time do I spend lifeless on my 140 mile journey? Roughly talking, one fatality happens for each 100,000,000 miles pushed. This ignores a number of issues (e.g., the truth that completely different speeds result in completely different fatality charges, which we’ll come again to later), but it surely’s adequate to present us a way of what’s occurring. My anticipated time spent lifeless on the journey is about

{frac{140 text{ miles}}{100,000,000 text{ miles per death}} cdot 50 text{ years per death} = 37 text{ minutes}}.

That’s quite a bit contemplating that, after I don’t die, my complete journey size comes out to about 2 hours and half-hour! 

However what occurs to my likelihood of dying if I drive quicker or slower? A considerably dated analysis from the University of Alabama offers a little bit of perception. To a primary approximation, for those who crash at a given pace, your likelihood of dying doubles every time you add 10 mph to that pace. That is to say that {Pr[text{die }midtext{ crash}]} doubles every time you enhance your pace by 10 mph. In fact, what we really care about is Pr[text{crash and die}] , which by Bayes’ rule is Pr[text{die }midtext{ crash}] cdot Pr[text{crash}]. It appears doubtless that each Pr[text{die }midtext{ crash}] and Pr[text{crash}] enhance as you drive quicker, however since I solely have knowledge on how the primary amount adjustments, I’ll ignore adjustments to the second.

Now let’s calculate the anticipated size of the journey (together with lifeless time) at completely different speeds. As a baseline, let’s suppose that our earlier evaluation (i.e., our estimate of 37 minutes lifeless) holds after we journey on the pace restrict of 65 mph. Now suppose we journey at 65 + x miles per hour for some x. Our precise time on the street is {frac{140 text{ miles}}{65 + x text{ miles per hour}}}, however our anticipated time spent lifeless is {37 cdot 2^{x / 10}} minutes. So our anticipated journey size, together with time spent lifeless, is:

{(60text{ minutes / hour}) cdot frac{140 text{ miles}}{65 + x text{ miles per hour}} + 37 cdot 2^{x / 10}} text{ minutes}

{= frac{8400}{65 + x} + 37 cdot 2^{x/10}} minutes.

Graphing the size of the journey (in minutes) as a operate of x offers:

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So the quickest pace to drive is… about 62 miles per hour.1

In fact, this mannequin bakes in your remaining life expectancy at 50 years. An odd characteristic of the mannequin is that, the older you get, the faster your journeys develop into, no less than in expectation, and the extra you must pace. Right here’s the identical graph for those who mannequin the price of dying as 25 years:

Now your optimum pace is nearer to 70 mph, nonetheless on the slower finish of what folks do on I-95. If you wish to rationalize the 85 mph speeds that the quickest drivers journey at, you would wish to cut back your estimated value of dying to a morbidly quick 5 years.

Now, for those who actually do wish to optimize how briskly you get locations, it’s price noting that this mannequin has a cheat code: completely different vehicles have very completely different fatality charges. (Though, sadly, they’re typically reported per registration 12 months reasonably than per mile). The trick to getting someplace quick isn’t to hurry, it’s to get a Volvo XC90.

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