Hack 61. Outsmart Superman

Lightning can strike twice in the same place, but it is very unlikely. The laws of probability allow us to calculate the likelihood of a series of rare occurrences happening all in a row.

Occasionally, we hear stories of highly unlikely events happening more than once to the same persona forest ranger who has been struck by lightning seven times, for example, or a New Jersey couple winning the lottery twice. When they appear in the news, these stories often include an interview with the local stats professor, who estimates the odds of such a thing happening.

The math for calculating the total likelihood of a series of events is fairly simple. The more difficult part is figuring good estimates for the probability of any single event happening once. Then, you simply multiply the individual probabilities together to get the total likelihood for the whole chain of weird happenings.

Lucky Lois Lane

To show the steps involved in calculating the likelihood of a whole series of events, I've chosen an example from classic literature. This series of rare events is described in the Lois Lane comic magazine #56, published by DC Comics in April of 1965. A common pattern in the stories involved Lois having some apparently supernatural experience that was hard to explain but, at the end of the story, turned out to have some simple explanation.

Lois Lane, now wife (but former girlfriend and number one fan) of the comic book hero Superman, was a very popular character in the line of DC comic books in the 1960s and 1970s. Among sophisticated comic aficionados, Lois Lane comics of that era are now enjoyed as examples of particularly strange comic writing. Lois tended to beat the odds almost on a daily basis. Her comics should be required reading in statistics courses.

One example of a strange experience that was then "explained" by Superman at the end of the story involved the application of a statistics hack. Lois is pretending to be telepathic so she can hang around mobster "Long Odds" Larkin and maybe get a scoop for her newspaper.

It works all too well, as she is kidnapped by Larkin and forced to provide him with "telepathic" information so he can commit crimes. Fortunately for Lois, and for the mobster, her blind guesses turn out to be correct and Larkin keeps her alive. Her guesses are so accurate that Lois comes to believe that she actually has psychic powers.

It turns out, according to Superman, who eventually rescues her, that Lois was just lucky! Very lucky. Astoundingly, incomprehensibly lucky. Even though the odds of Lois correctly making the lengthy series of correct predictions and accurate guesses were extremely slim, she just lucked out. Congrats, Lois!

Superman presents what he says are the odds for the fantastic feats that Lois performs, but the author of the story (anonymous) does not provide the calculations. Let's review the random guesses that Lois makes, do our own calculations, and check the Man of Steel's math. For determining the probability of this series of independent events, we will apply the multiplicative rule [Hack #25].

The Guesses

In the story, Lois correctly guessestotally at random, mind youthe following:

1. Which of five duplicate armored trucks is actually carrying Metro Bank cash

2. The combination to a safe that holds a large company's payroll funds

3. The unlisted phone number to the richest person in town

4. Under which of 20,000 trees a bank robber's loot is buried

She finally fails, after Superman has rescued her, to guess the number of jellybeans in a jar. As Superman explains to Ms. Lane that she is not psychic, he suggests that the odds of her making these four correct guesses by chance are 326,454,839,047 to 1, or 1 out of 326,454,839,048.

"I see, Superman!" she says. "I was lucky enough to hit that 'one chance'." "Yes," says Superman, "after all, someone always wins big lotteries, too" (or some such nonsense to that effect). That number calculated by Superman or his Supercomputer certainly is big, which seems right, but I don't think it is close to being correct. My guess is that this outcome is even more miraculous.

The Calculations

Let's work through our own calculations. For guesses 1 and 4, we can figure pretty close to the odds of guessing the answer to that problem independently. For guesses 2 and 3, we'll have to make some assumptions.

Here again are the guesses Lois made and real calculations of the odds for each one, taken by themselves.

The math involved here is the easy part for statisticians who are asked to produce statements of likelihood for a string of unlikely events. The hard part is determining the starting values, the pieces of the equations. As you see with our attempts to estimate how lucky Lois was, we will have to make some moderately wild, though reasonable, guesses to somehow know what the chances of any particular occurrence are. Statisticians can't really know the basic odds much of the time. They tend to focus on theoretical situations where the odds can be known, not real-life problems like those of Ms. Lane.

Guess 1

Which of five duplicate armored trucks is actually carrying Metro Bank cash? This is the easiest one. Five possibilities, one correct choice. The chances are 1 out of 5 or 1/5.

Guess 2

Lois guesses the combination to a safe that holds a large company's payroll funds. This is a real puzzler. Not only does Lois guess the five numbers that one should turn the dial to, but she also guesses that there is a sequence of five different numbers that must be used, and the directions that the wheel must be turned.

In the real world, there are a variety of different types of combination locks produced, so it is hard to know for sure what assumptions we should make about this problem. I've done a little research about safe cracking (for the sake of this hack, let's say) and learned a little about combination safes. Usually, there is a total of anywhere from one to eight numbers in a combination sequence. I'd guess that three or five numbers in a sequence is most common. The numbers on a dial can be any range of values, but 0 to 99 is common for larger safes, such as the payroll safe in the story.

So, for starters, let's say that she randomly picks between this safe having a three- or five-number combination. Chances for that guess are 1 out of 2, or 1/2. Say she randomly picks a number from 0 to 99 each time: 1 out of 100, or 1/100, for each number in the sequence. She also has to guess the starting direction. Let's say that most safes, 80 percent, start to the left, and only 20 percent, 1 out of 5, start to the right (which is her guess).

So far, so good. It gets very tricky here, though, because of the combination Lois actually suggests. She predicts "11 right...13 left...5 left...back to 8...forward to 15." This is a very odd combination. First, a combination is usually read in a different order: left 13, instead of 13 left. Second, what can it possibly mean to go left twice in a row! Surely you have to change direction of the dial to lock in each number in the sequence. After all, the dial passes over many numbers on its way left every time. How does it know whether to count each number it passes as a part of the combination sequence? I'm going to just pretend that the sequence is misreported slightly by the anonymous author; otherwise, I'd have to pause here in an endless loop of confusion, with my fingers over the keyboard, never able to continue.

Finally, why does Lois start saying "back" and "forward" instead of left and right? This just makes her directions unclear (perhaps to cover herself in case of failure?). Again, I'm going to assume she uses the terms to mean a change in direction, even though back probably means left and forward probably means right, which would just complicate things more. A conservative set of probabilities for this guess, then, is 1/2x1/5x1/100x1/100x1/100x1/100x1/100. That's 1 out of 100,000,000,000.

Guess 3

Lois also guesses the unlisted phone number to the richest person in town. There are a couple of ways to figure this.

First, if Lois were a bit naïve (and, no offense to Lois's fans, but I'm guessing she is), she might set only the parameters that the phone number had to have seven digits and not start with 0. Under these rules, there are 9,000,000 possible phone numbers. This assumes that we start with 10,000,000 possible seven-digit numbers (9,999,999 is the highest seven-digit number, plus add one for the number 0,000,000).

If we can't count any numbers that start with 0, that eliminates the number 0,000,000 and all six-digit or less numbers (there are 999,999 of those). That's an even million possibilities we can eliminate. So, under this scenario, Lois's chance of guessing the number would be 1 out of 9,000,000 or 1/9,000,000. Let's give Lois the benefit of the doubt for a second and imagine that she wouldn't guess her own phone number or other phone numbers she knows by heart. I'd guess there are maybe 10 of those. So, Lois would have 1 out of 8,999,990 to choose from.

A smarter Lois (let's say for the sake of argument) might know the particular exchanges in use in Metropolis, or those likely to be used for unlisted numbers, or for the rich part of town, or whatever. Back in the day, there was a small set of possibilities for the first three digits in a particular area code known as exchanges. A city the size of Metropolis might have fifty or so that were used most commonly, so she might choose from those. Under the "smart Lois" scenario, her odds improve considerably. Now, she might blindly guess out of 500,000 numbers, not 9,000,000. Her chances might have been 1 out of 500,000 or 1/500,000. My rough estimation of Lois's intelligence suggests that this scenario is not the most likely, but she is a reporter for a major metropolitan newspaper, so she may have this knowledge. Let's be charitable and go with it.

Guess 4

Finally, Lois guesses under which of "20,000" trees a bank robber's loot is buried. Like guess 1, this is also fairly easy to calculate. If there really are exactly 20,000 trees in the woods where the loot is buried (and this number is probably an estimate or rounded off), the chance of guessing correctly is 1 out of 20,000 or 1/20,000.

Final Probability

So, the chances of guessing correctly on these four problems in a row, giving Lois all sorts of benefits of the doubt for knowing all sorts of things about safes and telephone numbering systems, is 1/5x1/100,000,000,000x1/500,000x1/20,000. The chances of this sequence of lucky guesses occurring is, conservatively, 1 out of 5,000,000,000,000,000,000,000even more remarkable than the already hard to believe 1 out of 326,454,839,048.

"I see, Superman! I was lucky enough to hit that one chance," Lois concludes. Indeed. Of course, the odds were even worse that Superman would propose to Lois someday, and that happened. So, who am I to rain on Mr. and Mrs. Superman's parade?