Finding Pi by the Probability of Two Random Numbers being Co-Prime
Overview
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The probability that two random numbers (between zero and infinity) are co-prime (meaning they do not share any factors besides 1) is equal to six divided by pi squared (proved below). Therefore, theoretically, if an infinite number of random numbers (between zero and infinity) were generated, then the square root of six divided by the proportion of those that are co-prime is equal to pi. It is not obviously possible to actually do this an infinite number of times, so the process with two sample sizes: using 1000 random numbers by rolling dice, and a trillion random digits in Excel.
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Proof
Step 1: Determine the probability of two numbers sharing a common factor
Step 2: Determine the probability of two numbers being co-prime
Step 3: Find the sum of the series
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The series above is called the "Basel Problem". Euler was the first to solve it, however multiple in depth proofs can be found here. If these are too complicated, the video (right) gives an intuitive explanation of the Basel Problem using the inverse square law and light houses.
Proof Source and Credit: Cut-The-Knot.org
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Procedure by Hand
Automated Procedure
Troubleshooting Automation
Hypothesis 1:
