+95 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!
+9673 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 \(\dfrac{\text{Area}(P)}{\text{Area}(S)} = \dfrac{(5 - 1)^2}{(6 - 0)^2} = \dfrac49\).
+95 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!
