Why Not Infinitely Many More Rides?

Returning to the Disaster Transport ride problem: by flipping a coin after each ride of the roller coaster we’d decide whether to go around again. How many more times could I expect to ride? Using the letter k to represent the number of rides, and p(k) to represent the probability of getting that many rides, it’s a straightforward use of the formula for expectation value — the sum of all the possible outcomes times the probability of that particular outcome — to find the expected number of rides.

Where this gets to be a bit of a bother is that there are, properly speaking, infinitely many possible outcomes. There’s no reason, in theory, that a coin couldn’t come up tails every single time, and only the impatience of the Cedar Point management which would keep us from riding a million times, a billion times, an infinite number of times. Common sense tells us this can’t happen; the chance of getting a billion tails in a row is just impossibly tiny, but, how do we know all these outcomes that are incredibly unlikely don’t add up to something moderately likely? It happens in integral calculus all the time that a huge enough pile of tiny things adds up to a moderate thing, so why not here?

Just One More Ride?

Given that we know the chance of getting any arbitrary number — let’s say k, because that’s a good arbitrary number — of rides in a row on Disaster Transport, using the scheme where we re-ride if the flipped coin comes up tails and stop if it comes up heads, the natural follow-up to me is: how many more rides can we expect? It’s more likely that we’d get one more ride than two, two more rides than three, three more rides than four; there’s a tiny chance we might get ten more rides; there’s a real if vanishingly tiny chance we’d get a million more rides, if Cedar Point didn’t throw us out of the park and tear the roller coaster down first.

How Many Last Rides?

So our scheme for getting a last ride in on Disaster Transport without knowing in advance it was our last ride was to flip a coin after each ride, and then re-ride if the coin came up tails. (Maybe it was heads. It doesn’t matter, since we’re supposing the coin is equally likely to come up heads as tails.) The obvious question is, how many times could we expect to ride? Or put another way, how many times in a row could I expect a flipped coin to come up tails, before the first time that it came up heads? The probability tool used here is called the geometric distribution.

The Last Ride Of A Roller Coaster

Cedar Point amusement park, in Sandusky, Ohio, built in the mid-1980s a bobsled-style roller coaster named Avalanche Run, because it was the mid-1980s and bobsled-style roller coasters seemed like a good idea. My home amusement park, Great Adventure, had something called the Sarajevo Bobsled opened in that time because back then Sarajevo was thought to be a pretty good city apart from that unpleasantness seventy years before. But Cedar Point’s bobsled roller coaster had a longer existence than Great Adventure’s, and around 1990, it was rebuilt to something newer and more exciting, with a building enclosing it and a whole backstory behind the ride.

A Brief Word for the Comic Pages

There’s legitimate mathematical content linked from here, but mostly, I want to promote what seems to be a little-known comic strip that’s working very hard at making me love it. Part of that work has been in producing a couple of mathematics-oriented strips. Grant Snider’s Incidental Comics, part of the gocomics.com comics empire, is a roughly twice-a-week strip filling the page with lots of detail humor. It’s the sort of comic strips which assumes you will remember Ludwig Miles van der Rohe’s Farnsworth House. (However, I did see a Lego block version of the Farnsworth House in Barnes and Noble the other day, so maybe Miles van der Rohe has gone and become all trendy while I wasn’t looking.

Relevant to the nominal base for this little blog, though, is that Snider has posted a few comics based on mathematics jokes. The most recent is that from January 23, titled “Axes of Evil”, and mixes descriptive statistics with horror that is somehow not associated with calculating standard deviations. A little farther back is the December 12, 2011, strip, titled “Function World”, which adapts graphs of some popular functions, such as hyperbolas, the natural logarithm, and the inverse cosine (which is not actually popular, but don’t tell it) into amusement park rides. Do enjoy.

I am not certain how far in the archives people who haven’t got gocomics.com accounts can go before they’re nagged into getting gocomics.com accounts.