Math Forum: Probability and the Problem of Points: Summary

Summary: A Great Moment in Mathematics

Table of Contents:

The Bridges of Konigsberg
· Euler's Solution
· Solution, problem 1
· Solution, problem 2
· Solution, problem 4
· Solution, problem 5

The Value of Pi
· A Chronological Table of Values
· Squaring the Circle

Prime Numbers
· Finding Prime Numbers

Famous Paradoxes
· Zeno's Paradox
· Cantor's Infinities
· Cantor's Infinities, Page 2

The Problem of Points
· Pascal's Generalization
· Summary and Problems
· Solution, Problem 1
· Solution, Problem 2

Proof of the Pythagorean Theorem

Proof that e is Irrational

Book Reviews

References

Links

The logic displayed on the previous two pages may seem simple, but Pascal and Fermat had therein grasped a very important concept that, though intuitive to us, was revolutionary in 1654. This was the idea of equally probable outcomes. They realized that the 'probability' of something happening could be computed by enumerating the number of equally likely ways it could occur, and dividing this by the total number of possible outcomes of the given situation. This is what Fermat did when he figured out the 16 different (equally likely) results of tossing a coin 4 times, and then counted the ones which would result in a win for him. Again, this may seem obvious to you, but many great mathematicians of the 15th and 16th century gave 'solutions' to the problem of points that were just plain wrong. Something along these lines has happened in the last twenty years with the Monty Hall Problem.

Problems:

Problem 1

Suppose Pascal had been winning 9 to 6 when they he and Fermat were interrupted. How should the 100 Francs be divided?

Solution

Problem 2

Suppose you had come upon them earlier and noticed that they were tied 5 to 5. You leave and come back 5 flips of the coin later. What's the probability of Pascal being ahead 9 to 6?

Solution

to Pascal's Generalization
to a Proof of the Pythagorean Theorem

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August, 1998