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Using Geometry with Probability

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

Thanks for the help!

Apr 8, 2024

#1
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Suppose X and Y are random variables which follow the continuous uniform distribution on [0, 6]. These random variables will represent the points where we break the stick at.

The probability in question is simply expressible as $$\mathcal P (\min(X, Y) \leq 5\text{ and }\max(X, Y) \geq 1\text{ and }|X - Y| \leq 5)$$.

Let $$S = \{(x, y): 0\leq x \leq 6, 0 \leq y \leq 6\}$$ and $$P = \{ (x, y) \in S\mid\min(x, y) \leq 5\text{ and }\max(X, Y) \geq 1\text{ and }|X - Y| \leq 5\}$$. The required probability is $$\dfrac{\text{Area}(P)}{\text{Area}(S)}$$. The following is a graph of P:

Now, the probability is $$\dfrac{\text{Area}(P)}{\text{Area}(S)} = \dfrac{(5 - 1)^2}{(6 - 0)^2} = \dfrac49$$.

Apr 8, 2024
#2
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I think a slight problem to your solution is that you are saying our (X,Y) can be equal to 5 but the quesiton doesn't want our pieces to even equal to five. Would this change your solution? I haven't entered anything. I wanted to check before doing any submitting.

Thanks for the help!

Apr 8, 2024