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How Isaac Newton got a famous dice puzzle both right and wrong

Sir Isaac Newton is known for many advances in many different fields of knowledge. He formulated a theory of gravity after a famous encounter with an apple, deriving a lot of information about our Solar System. He made seminal contributions to optics, and he was heavily involved with the development of calculus. One area he didn’t overly engage with was the then-essentially unknown field of probability, but he was involved with a famous probability problem. Newton’s solution was ingenious, and both right and wrong at the same time, and it’s a fascinating look how we approach problems.

In the late 17th century, Newton shared a series of correspondence with Samuel Pepys, the famous diarist and member of the Royal Society of London (he served as President from 1684-86). In 1693, Pepys wrote to Newton asking for his opinion on a wager that he wanted to make. Which of the following three propositions has the greatest chance of success: A, six fair dice are tossed independently and at least one ‘6’ appears; B, 12 fair dice are tossed independently and at least two ‘6’s appear; or C, 18 fair dice are tossed independently and at least three ‘6’s appear?

It’s a simple enough question, but surprisingly complex to answer. Think about your own gut reaction to this – some people think it is either A, B or C. In addition, some think it doesn’t matter in terms of probability – how can the probability change when the odds of rolling a six on any fair dice is always 1/6? Pepys’ own belief was that C would be the most likely outcome, followed by B and then A, but Newton proved the opposite – A is the most likely outcome of the dice rolls. I’m not going to run through all the maths here (you can see the process, obtained by applying the binominal distribution, here), but Newton demonstrated that A has around a 66.5% chance of occurring. For B, this falls to 61.9%, and the likelihood of C is 59.7%.

 It’s a simple enough question, but surprisingly complex to answer

Until the 19th century, probability wasn’t really a thing (and it was only generally applied to gambling when it appeared), meaning Newton didn’t have access to the binominal distribution – instead, he worked out everything with a detailed series of tables, which would enable the reader to follow his maths. It used a set of first principles, meaning you could plug in whatever values you wanted to get your own results, and it’s really impressive work.

But, if these numbers are right, where does the error creep in? A paper published in 2006 by Stephen M. Stigler highlighted a major problem made in Newton’s reasoning – he approached the problem imagining that you needed to throw one ‘6’ for each block of six dice, which is obviously far likelier to do once, and so he said A was likeliest. But this doesn’t quite line up with Pepys’ original problem. If you rolled three ‘6’s immediately and then 15 other dice without a ‘6’, that meets the terms of option C but would be considered a fail by Newton’s reasoning. Newton was conscious that there was the potential for slippage around defining the terms of the problem, and he told Pepys as much in a letter from December 1693, but the probabilistic thinking wasn’t there for him to know exactly why.

He knew he was right, but he was wrong about why he was right, which makes his conclusion partially wrong too

We wind up here, then with a very interesting situation – the gambling advice that Newton gave was correct, but the logical argument underpinning it was not. He knew he was right, but he was wrong about why he was right, which makes his conclusion partially wrong too. I like the Newton-Pepys problem because it demonstrates that being able to do the maths, as it were, is only part of the question – how you understand and engage with a question also affects the results you’ll get, and the conclusions you’ll draw. To this day, there are still people who argue with the maths at the heart of this problem, indicating just how important perception can be too.

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