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

apples1ices Sep 29, 2023

#1**-1 **

We can solve this problem using geometric probability. Consider the following diagram:

[asy] size(100); pair A=(0,0), B=(6,0), C=(0,5), D=(6,5); draw(A--B--C--D--cycle); draw(B--((2,0)+(0,4))); draw(B--((4,0)+(0,3))); draw(B--((6,0)+(0,2)));

label("A", A, SW); label("B", B, SE); label("C", C, SW); label("D", D, SE); label("(2,0)", (2,0), SW); label("(4,0)", (4,0), SW); label("(6,0)", (6,0), SW); label("(0,4)", (0,4), NW); label("(0,3)", (0,3), NW); label("(0,2)", (0,2), NW); [/asy]

The three points B, (2,0), and (4,0) divide the square ABCD into three regions, each of which corresponds to one possible outcome of the experiment. The shaded region in the diagram corresponds to the outcome that all three pieces are shorter than 5 units.

To calculate the probability of this outcome, we need to find the area of the shaded region and divide it by the area of the entire square. The area of the shaded region is a right triangle with base 2 and height 4, so its area is 21⋅2⋅4=4. The area of the entire square is 6⋅5=30, so the probability that all three pieces are shorter than 5 units is 4/30=2/15.

bingboy Sep 29, 2023

#3**+1 **

Bingboy, have you actually thought about whether this answer is reasonable? You should always to that.

If you do not have enough knowledge to know then why use a computer program or AI to answer, just leave it to someone who has some idea what they are talking about.

Melody
Sep 29, 2023

#4**0 **

Almost all of the non-CPhill answers are ChatGPT generated, especially if they are from the "regular askers" such as bingboy, maximum, sandwich, etc...

I mean seriously! Melody and CPhill are giving you free help on your math problems. The least you could do is **not** post blatantly incorrect spam.

plaintainmountain
Sep 30, 2023

#2**+2 **

Let one part be x units long. Let the next part be y-x units long and let the last bit be 6-y units long

when you add these together you get x + (y-x) + (6-y) = 6 units.

Now graph this on a number plane

0 < x < 6

0< y-x < 6

0 < 6-y <6

The area inside this shape represents the total possibility of the stick breaking anywhere. I mean, no matter where the stick is broken it will be represented by a point inside this region.

You want all three bits to be less than 5.

So now you have another smaller regaion to draw,

0 < x < 5

0< y-x < 5

0 < 6-y <5

So the probability that you want is the little area divided by the big area.

Let me know how you go. Maybe take a photo of your graph and post it here.

Melody Sep 29, 2023