Smaller Random Squares

Probability Level 1

A factory manufactures wooden squares of 4 square centimeters. Due to some manufacturing defect, the side length of the square varies uniformly between $1\text{ cm}$ and $3\text{ cm}.$

Manager Bob wants to know how frequently the squares are smaller, i.e. less than $4\text{ cm}^2$ in area. He made the following observations:

The area of the square to be manufactured next is a random quantity between $1\text{ cm}^2$ and $9\text{ cm}^2.$
So, the probability that the area of the next square is less than $4\text{ cm}^2$ is $\frac{3}{8}.$

Is Bob correct?

Yes No

2 solutions

Michael N.
Nov 14, 2017

The area of the square is not randomly distributed, the sidelength is. The sides of the square remain porportionate to each other. Since the side length varies between 1cm and 3cm, and a square with an area of 4cm^2 will have a sidelength of 2cm, one can assume that approximately one half of the squares will have an area less than 4 square centimeters.

Andrei Stephenson
Nov 18, 2017

To add to Michael's response, assumption 1 is incorrect as Michael pointed out in the first part. Secondly, since no prior information was given about the likelihood/probability of each type of defect to occur it is reasonable to assume that they are equally probably. In which case as Michael points out would imply that 50% of the time the area would be below 4 cm^2.

Smaller Random Squares

2 solutions

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