Question

A point is chosen at random on a line segment of length L. Interpret this statement, and find the probability that the ratio of the shorter to the longer segment is less than 1/4.

Answer 1

Answer:

Probability=2/5

Step-by-step explanation:

Let the line segment be AB of uniform length and x is distance

so

$P(\frac{min(x,L-x)}{max(x,L-x)} )$

Case 1 min(x,L-x) then

$=\frac{x}{L-x}\leq 1/4\\x$

Case 2 min(x,L-x)=L-x then

$=\frac{L-x}{x}\frac{4L}{5}$

So Probability

$P=\int\limits^{L/5}_0 {f(x)} \, dx+\int\limits^{L}_{4L/5} {f(x)} \, dx\\ P=\int\limits^{L/5}_0 {\frac{1}{L} } \, dx+\int\limits^L_{4L/5} {\frac{1}{L} } \, dx\\ P=\frac{1}{L}(\frac{L}{5} -0+L-\frac{4L}{5} )\\ P=\frac{1}{L} (2L/5)\\P=\frac{2}{5}$