Bertrand’s Paradox
There is a circle that has a radius of 1 cm long given. We draw an equilateral triangle in it with the edge √ 3…
There is a circle that has a radius of 1 cm long given. We draw an equilateral triangle in it with the edge √ 3. Determine the possibility of randomly choosing a chord which is greater than √3. It might seem easy to solve at first but it is actually a paradox which is found by Joseph Bertrand, a mathematician. As it is a paradox, it has 3 different answers.
The first possible answer is 1/3. As you see beneath this paragraph, with the help of symmetry we can consider that one of the vertices of the possible chords is one of the vertices of the triangle.
Then it becomes clear that if the other vertice of the chord is between the 2 edges then it is greater than √3. So one-third of the chords are greater than square root 3. Then the probability is 1/3.
The second answer is 1/2. It comes from the fact that if the midpoint of the chord is chosen randomly then the chord is also chosen randomly. So we can choose the midpoint of the chord instead of choosing the chord. The distance between the midpoint of each edge and the centre is 1/2. If the distance between the centre and the midpoint of the chord is greater than 1/2, then that chord is less than √3. If the distance between the centre and the midpoint of the chord is less than 1/2, then that chord is greater than √3. Then the probability is 1/2.
The third possible answer is 1/4 which comes from nearly the same idea in the second solution. A randomly chosen midpoint indicates a randomly chosen chord. As a result, we want to choose the midpoint inside the incircle of the triangle. Then the probability becomes the ratio of the incircle and the main circle which is 1/4.
Joseph Louis François Bertrand was a professor at the École Polytechnique and Collège de France. He was a member of the Paris Academy of Sciences. His father, Alexandre Jacques François Bertrand, was a physician and his brother Alexandre Bertrand was an archaeologist. His father died when he was nine years old. He attended the course of École Polytechnique as an auditor when he was eleven years old. He obtained two bachelor degrees, a licence and a PhD with a thesis on the mathematical theory of electricity, from age eleven to seventeen.
He conjectured that there is at least one prime between n and 2n − 2 for every n > 3, in 1845. Chebyshev proved this conjecture in 1850. Now it is called Bertrand’s Postulate. He was also famous for his paradoxes in probability and game theory.
He translated Karl Freidrich Gauss’ work on the theory of errors and the method of least squares into French.
His book “Thermodynamique” points out in Chapter XII, that thermodynamic entropy and temperature are only defined for reversible processes. He was one of the first people to point this out.
He was elected a foreign member of the Royal Swedish Academy of Sciences, in 1858.
Resources:
- Bertrand’s Paradox. MIT. Web. 28.11.2019
- Bertrand’s paradox. bertrands-paradox. Web. 28.11.2019
- Paradoxes in Probability. Brilliant. Web. 27.11.2019
- Bertrand’s Postulate. WolframMathWorld. Web. 01.12.2019
- Bertrand, Joseph Louis François. encyclopedia. Web. 01.12.2019
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