Vsauce! Kevin here, with historical proof that Isaac Newton was wrong at least once
in his storied career. He seems to have been right about… gravity… and calculus…
and 58,000 other things that redefined how humans thought about math and science. But
here’s what Newton couldn’t quite handle:
DICE.
Some of ye are gonna think I’m wrong. And others will agree that Newton was wrong but
ye will think that it doesn't even matter. This will be fun.
Let’s start by talkin’ about Pepys.
Samuel Pepys gained fame as England’s Chief Secretary to the Admiralty and eventually
became the 6th president of the Royal Society of London -- the oldest national scientific
institution in the world. But now he’s most famous for his letters and meticulous diaries.
In 1693, Pepys sent a dice gambling problem to one of the most brilliant mathematical
minds alive: Sir Isaac Newton.
It’s the only time Newton seems to have worked on probability -- and even though he
was mostly right, he was still a little bit wrong. Here’s how.
Pepys made a wager with a man named John Smith over which of the following scenarios was
the most likely to occur -- or if they were all equal:
A - has 6 dice in a box, with which he is to fling a 6;
B - has in another box 12 dice, with which he is to fling 2 sixes;
C - has in another box 18 dice, with which he is to fling 3 sixes.
Q - Whether B and C have not as easy a task as A at even luck?
Pepys’s problem isn’t hard to figure out… until it is. On the surface, is it easier
to get one six in 6 dice, or three sixes in 18 dice? Or… is there no difference at all
in terms of probability? Pepys posited that C, with three 6s in 18 dice, was the most
probable. More dice, better odds.
Hey... What are we even doing with YE DICE ?
We’re using standard 6-sided dice with each side equally likely as an outcome. No funny
business here. The odds of getting a 6 on each individual roll are a simple 1 in 6.
Now we just need to “fling” them.
WAIT -- we’ve got a language problem. Is the bet here to find out whether we’re flinging
a 6, or does that imply at least one 6? Because exactly one 6 and at least one 6 are two different
things, and that would apply to B and C, too. It’s a critical point in figuring out exactly
what our odds are and whether there’s any difference. As they exchanged letters, Newton
interpreted Pepys’ problem as being satisfied with at least the minimum number of sixes.
And while Pepys predicted that the best shot to win was rolling at least three 6s in a
fling of 18 dice, Newton proved it was the opposite: A is most likely, B is next most
likely, and “In ye third case, ye value will be found still less.”
And I know that people don’t believe this. There are articles written to this day, 300
years later, claiming that the probability does not change when ye add dice. After all,
every die has that 1 in 6 chance of coming up 6… and a set of 6 dice is really just
6 individual dice. 12 and 18? That’s just two and three sets of 6. But they are wrong!
And I’ll show ye why.
It’s time to prove this by flinging some dice. Mathematically.
All the possible probabilities of an event need to add up to 1 -- so in a coin flip,
there’s a .5 probability of heads and a .5 probability of tails -- and the probability
of one of those two outcomes occurring is .5 + .5 = 1, because one of those two things
is certain to happen. It’s the same with our dice game.
So to find out the probability of something happening, it’s usually best to calculate
the probability of that thing not happening and then subtracting it from 1. Those are
the only two outcomes here, it happens or it doesn’t, so that calculation “everything
minus failure” leaves us with the probability of success.
The probability of rolling a 6 is p... Hold on I gotta find my marker. Probability of
finder my marker -- 100%! The probability of rolling a 6 is p. So p = ⅙. The probability
of not rolling a 6 is (1 - p), so the probability of not rolling a 6 with 6 dice -- our A scenario
-- is 1 - (1-p)^6.
Plug in the ⅙ value for p and the probability of rolling at least one 6 with just 6 dice
= .6651…
We do the same thing for B. It’s not any more sophisticated, there are just more terms…
and a little extra step, since rolling only one 6 in 12 dice is now considered a failure,
too. So the probability of B equals:
B = 1 - (1-p)^12 - (12/1)p(1 - (1-p)^11
That simplifies to: 1 - (1+11p)(1-p)^11
… and when we put in our ⅙ value again, the probability of succeeding in scenario
B is .6187…
It’s about 5 percentage points less than the odds of winning with only 6 dice. And
that’s actually about a 7% swing in outcomes.
And like Newton said, C is even worse: “ye value will be found still less” turned out
to be about .5973. But Newton didn’t actually calculate the
third scenario. He had to calculate all of this math by hand -- not using the simple
probability notation we just used -- and he actually made a really detailed series of
tables that Pepys could use to do the calculations himself. Newton’s tables used first principles
-- basic knowledge rather than complicated, specialized knowledge -- so Pepys could confirm
the calculations like this and plug in his own values. So for example, the probability
of at least one “6” in a toss of 6 dice is obtained from column F by dividing row
9, “outcomes with at least one ‘6’” by row 2, “total outcomes for the dice thrown.”
It’s simple, elegant genius. I, and ye, can do it a lot more simply and crunch the
numbers on a calculator, or run a simulation of, say, 100,000 trials of 18 dice-rolls,
in just a few seconds to confirm the result. But if ye want tables like Pepys, ye got tables.
Isaac Newton didn’t have those computational tools, and to be honest, he just wasn’t
that familiar with probability. No one in the 17th century was. And that’s why he
got the numbers right, but he got the reasoning wrong.
In 2006, Stephen Stigler, a professor of statistics
at the University of Chicago, detailed a serious
flaw in Newton’s logic. As the correspondence evolved, Newton assigned Scenario A to a fictional
man named Peter and Scenario B to James. He explained that Peter only needed one 6 in
his fling of 6 dice, while James needed a 6 in each fling of 6 dice -- so, it was easier
for Peter to win. Same with the unnamed Player C, who’s flinging 18 dice. Newton completely
ignored the possibility of getting more than one 6 in each fling, and that’s just…
a different problem than the one we’re dealing with here.
If that seems like a minor detail, it really isn’t. Yeah, his explanation was wrong but
his answer was right. And that seems to be enough for most people. Almost no one talks
about how his conclusion was wrong. And that’s what’s amazing about this problem to me.
Does it matter if ye are wrong about why ye are right?
On December 23, 1693, in his third letter on Pepys’ dice problem, Newton’s explanation
conflated that original language problem I pointed out -- whether we’re hoping to get
exactly a number of 6s, or at least a certain number.
He was confused. He knew the math and he nailed it, despite probabilistic thinking not even
really being a thing in his era… but his reasoning couldn’t quite catch up to the
numbers.
That’s okay.
And for centuries, we kinda let that go. Part of it was that it took a long time to develop
enough knowledge to know why Newton’s explanation wasn’t exactly right. Probability seems
to have evolved from wanting a better understanding
of gambling, and the numbers were accurate.
And part of it was that… he’s Isaac Newton. No one’s making a career off disputing the
details of a man on humanity’s Mount Rushmore of math and science. Except Stigler. And I
guess… me.
But the Newton-Pepys problem shows us how numbers can’t just exist on their own. They
work hand-in-hand with our understanding of the problems they’re applied to. And really,
Newton’s solution is evidence that Occam’s Razor is right: that the simplest method is
probably the right one, and the simpler sequence of events is probably going to work more often.
We use that mindset to develop and improve everything from manufacturing and assembly
lines to coding software for sustainability. We need it to progress.
The important thing is someone needs to get complicated about getting simple to prevent
seemingly simple things from suddenly getting… complicated. Which makes things simpler for
us so that we can move on to things that are more complicated.
And as always -- thanks for watching.