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### Course: Wireless Philosophy > Unit 2

Lesson 8: Probability# Bertrand's Paradox

In this Wireless Philosophy video, Jonathan Weisberg (University of Toronto) explains Bertrand's Paradox, a famous paradox in probability theory. Beginning with the square factory example, he talks about how Bertrand's Paradox reveals a puzzling problem for the principle of indifference and the implications of this paradox for scientific reasoning.

## Want to join the conversation?

- Sticking my head out and I am aware I may be absolutely wrong, but I think this approach may solve the paradox:

In my most humble opinion, the paradox arises due to, for instance, comparing apples with oranges, since we are comparing values in a linear range with values of a quadratic function. To compare like with like, think slope (1st derivative) of the quadratic curve for corresponding area values should be used.

The 1st derivative (slope) of the equation y = x^2 can be calculated as 2x. The corresponding values at relevant intervals can be calculated as follows:

for length x = 1, area = 1, and 2x = 2;

for length x = 2, area = 4, and 2x = 4;

for length x = 3, area = 9, and 2x = 6;

A: corresponding 2x range for interval x = 1 to 2, is 4 - 2 = 2;

B: corresponding 2x range for interval x = 1 to 3, is 6 - 2 = 4;

Therefore, corresponding probability of A / B can be calculated as 2 / 4 = 1 / 2 = same as the probability calculated for corresponding length range.(6 votes)- Apart from the reason that it solves the paradox, why use the slope?(1 vote)

- The reason why the probability considering the length is different from the probability considering the area is that, with the square, the length is sufficient to define it in all the possible squares with lengths 1-3ft. But with the area, it is not the case. The area that ranges from 1-9 defines all the possible shapes having that area, not just squares. So by switching to the area, we're losing a condition for a shape to be a square, hence, the possibility of 3/8 isn't the probability of squares out of all squares like what 1/2 is.(3 votes)