A stick has a length of 5 units. The stick is then broken at two points, chosen at random. What is the probability that all three resulting pieces are SHORTER THAN THREE UNITS.


Many people have posted 4/25, 16/25 but those are wrong,  PLS HELP.

 Apr 15, 2022

use the logic from previous answers to answer this, it shouldn't be that hard

 Apr 15, 2022

It's super hard.  I've tried that multiple times, but I just get so stuck.  Pls help me!!  

Guest Apr 15, 2022

Can somebody pls try and answer this one?

 Apr 15, 2022

CPhill, pls help me!!!!!!!!

 Apr 15, 2022

I'm not sure if my logic is right, but anyways...


Think about it this way... You break it into 2 points initially. There are 2 cases. 1 stick is longer than 3 units (1) or both sticks are shorter than 3 units (2). 



The probability of (2) happening is \(1 \over 5\)


Once you split it into 2 sticks, each less than 3, you are guaranteed to get a 3rd stick that is also less than 3.


It is the region that is on the line \(x+y=5\), but bounded by the inequalities \(x \leq 3\) and \(y \leq 3\).


Here is a graph if you would like a more "visual" proof: https://www.desmos.com/calculator/bqrv2fkhhz



The probability of (1) happening is very complicated because the probability depends on where the "breaking point" was. 


The best-case scenario is when you have 2 sticks that are split into a stick with length 3 and a stick with length 2. 


Here, you have a \(3 \over 5\) chance of success, because if you break it anywhere in the segment with a length of 3, you split it into 2 sticks each with a length less than 3


However, the worst-case scenario is when you split it into lengths 5 and 0 (or smth very close to it). 


Here, you only have a \(1 \over 5\) chance of splitting the stick into 2 lengths of 3, as shown in (2). 


Because the probability depends, you take the average, which is \(2 \over 5\) (I think...)


Thus, the probability is \({1 \over 5 }+ {2 \over 5} = \color{brown}\boxed{3 \over 5}\)

 Apr 15, 2022
edited by BuilderBoi  Apr 15, 2022

Please refer to my answer here: https://web2.0calc.com/questions/probability_11169#r1

The answer is \(\dfrac{13}{50}\).

 Apr 16, 2022

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