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Do You Believe...

What is the probability that the equation x2+2bx+c = 0 has real roots?

A Warning: This problem is posed because it seems to be a good example of a case where the experimental probability does not approximate the theoretical probability.


Source: Wish I could remember the source of this problem...?

Hint: Think "geometric probability" where one uses ratios of areas to gain probabilities. But, what does area have to do with this problem?

It is okay to try some random values b and c, but this experimental approach could lead you astray...


Solution Commentary: The following is a suggested solution, noting upfront that numerous people have disagreed with its premise and approach.

Draw an axis system, with the horizontal axis being b-values and the vertical axis being c-values. Draw a B-square centered at the origin with side lengths 2B, i.e. its sidea pass through the points (0,B), (B,0), (0,-B), and (-B,0).

Now, suppose that the values b and c are chosen randomly from within this square. All possible points b and c can be found when the value B approaches infinity.

The original equation has real roots when the discriminant 4b2-4c ≥ 0 or b2-c ≥ 0. But, this statement can be visualized as a parabola section within the B-square, using c = b2. Shade the area contained within the parabolic section within the B-square, as that represents the condition b2-c ≥ 0. Then, for random points (b,c) within the B-square, the probability that the quadratic x2+2bx+c = 0 has real roots is the ratio of the area within this parabolic section to the area of the B-square.

Using integration, we learn that the area within the parabolic section (check it out!) is (4/3)B3/2, and the area of the B-square is 4B2.

Thus, the desired probability for random (b,c) within the B-square is [4B2-(4/3)B3/2]/[4B2] = 1-(1/3)B-1/2.

For example, if B = 4, the probability of real roots is 5/6...remember, the random values b and c are determined by random points in the B-square.

Finally, now let B go to infinity, in order to pick up all possible random values of b amd c. Then, we see that the probability that the equation x2+2bx+c = 0 has real roots is 1.

Do you believe this?