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.
