1. ## finite probability question

A point X is chosen at random on a line sement AB. (a)show that the probability that the ratio of lengths AX/BX is smaller than a is = to a/(1+a). and (b) show that the probability that the ration of the length of the shorter segment to that of the longer is less than 1/3 is = to 1/2.

2. Originally Posted by lauren2988
A point X is chosen at random on a line sement AB. (a)show that the probability that the ratio of lengths AX/BX is smaller than a is = to a/(1+a). and (b) show that the probability that the ration of the length of the shorter segment to that of the longer is less than 1/3 is = to 1/2.
You may as well assume that the line segment is of length $\displaystyle 1$, and that you choose a point $\displaystyle x\sim U(0,1)$. Now the first part of the question becomes:

find $\displaystyle p(x/(1-x)<a)$ give $\displaystyle x\sim U(0,1)$.

You do this by rearranging the condition $\displaystyle x/(1-x)<a$ into the form $\displaystyle x< ...$ where $\displaystyle ...$ is an expression in $\displaystyle a$ only.

CB

3. Originally Posted by lauren2988
A point X is chosen at random on a line sement AB. (a)show that the probability that the ratio of lengths AX/BX is smaller than a is = to a/(1+a). and (b) show that the probability that the ration of the length of the shorter segment to that of the longer is less than 1/3 is = to 1/2.
To do the second part we are being asked (same notation as before) to compute the probability that $\displaystyle x<1/4$ or $\displaystyle x>3/4$

CB