A stick is broken at two points, chosen at random. If the length of the stick is 6 then what is the probability that all three resulting pieces are shorter than 5 units?

Thanks in advance

Currykirby Apr 7, 2023

edited by
Guest
Apr 7, 2023

#1**+1 **

Let us label the broken pieces as $x$, $y$, and $z$, where $x$ is the length of the piece closest to the left end of the stick, $y$ is the length of the middle piece, and $z$ is the length of the piece closest to the right end of the stick. Without loss of generality, we can assume that $x\leq y\leq z$.

The total sample space of all possible ways the stick can be broken is determined by choosing two points on a stick of length 6. There are a total of ${{6}\choose{2}}=15$ ways to do this.

Now we need to find the subset of this sample space in which all three pieces are shorter than 5 units. For this to happen, we must have $x < 5$, $y < 5-x$, and $z < 6-x-y$. Note that $z$ can be at most 6 minus the sum of the lengths of the other two pieces, because the pieces cannot overlap.

We can graph the region of the $(x,y)$ plane where these three conditions are all satisfied. This region is a triangle with vertices at $(0,0)$, $(5,0)$, and $(2.5,2.5)$.

[asy] size(8cm); draw((0,0)--(5,0)--(2.5,2.5)--cycle); fill((0,0)--(5,0)--(2.5,2.5)--cycle,gray(0.7)); draw((5,0)--(6,0)); draw((2.5,2.5)--(6,2.5)); draw((0,0)--(0,6)); draw((0,6)--(6,6)); draw((0,0)--(6,6),dashed); label("$x$", (6,0), S); label("$y$", (0,6), W); label("$(0,0)$", (0,0), SW); label("$(5,0)$", (5,0), S); label("$(2.5,2.5)$", (2.5,2.5), NE); [/asy]

The area of this triangle is $\frac{1}{2}(5)(2.5)=6.25$, so the probability that all three resulting pieces are shorter than 5 units is $\frac{6.25}{15}=\boxed{\frac{5}{12}}$.

Guest Apr 7, 2023

#3**+2 **

Let the thr peices be these lengths x, 6-y and y-x

if you add those together you get a total length of 6 units

You know that each of those must be between 0 and 6 units long and that y>x

Graph the intersection of all those on a nu7mber plane. The area represents the sample space.

now graph where all the pieces are <5 units Find the area of that bit. That is the desired event area

now put the desired event area over the sample space area and you have your answer.

Melody Apr 8, 2023