newton-pepys-problem

A console application that tests a famous dice probability problem known as the Newton–Pepys Problem.

The Problem

Would it be more likely to roll one six in 6 dice, two sixes in 12 dice, or three sixes in 18 dice?

For purposes of this test, we will assume that groups that land on more than 1 six per set are also considered "successful", although the original problem posed to Newton and Pepys by a school teacher named John Smith was ambiguous as to whether throwing more than 1 six per set of six dice should be considered "successful".

The Test

1. Throw 6 dice counting how many land on the "6" face
2. Throw 12 dice and count how many land on the "6" face
3. Throw 18 dice and count how many land on the "6" face

Surprisingly, the probability is different for each of these.

• 6 dice = 66.51% chance of at least one six
• 12 dice = 61.87% chance of at least two sixes
• 18 dice = 59.73% chance of at least 3 sixes

In my approach, I throw all throws in sets of 6, rather than 12/18/etc. at the time. This ends in the same result as if we were to do it the other way.

For more history and a mathmatical explanation, visit the wikipedia page for this problem.

Running this test gives these results:

Rolled all sets 1,000,000 times

• Percentage of sets containing 1 throw that had at least 1 die landing on 6: 66.5682%
• Percentage of sets containing 2 throws that had at least 2 die landing on 6: 61.9443%
• Percentage of sets containing 3 throws that had at least 3 die landing on 6: 59.7407%