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# PLS HALP

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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

#1
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use the logic from previous answers to answer this, it shouldn't be that hard

Apr 15, 2022
#2
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It's super hard.  I've tried that multiple times, but I just get so stuck.  Pls help me!!

Guest Apr 15, 2022
#3
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Can somebody pls try and answer this one?

Apr 15, 2022
#4
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CPhill, pls help me!!!!!!!!

Apr 15, 2022
#5
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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
#6
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The answer is $$\dfrac{13}{50}$$.