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A stick is broken at two points, chosen at random. If the length of the stick is 6 then what is the probability that all three resulting pieces are shorter than 2 units?

 Mar 15, 2023
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Let the stick be a line segment of length 6, and let X and Y be the points where the stick is broken, chosen at random. We want to find the probability that all three resulting pieces are shorter than 2 units. 

First, we can assume without loss of generality that X < Y. Then the three pieces have lengths X, Y-X, and 6-Y. 

For all three pieces to be shorter than 2 units, we need:

- X < 2
- Y - X < 2
- 6 - Y < 2

Simplifying these inequalities, we get:

- X < 2
- Y < X + 2
- Y > 4

We can visualize these inequalities in the following region:

```
      Y
      |\
      | \
      |  \
      |   \
______|____\______ X
      0   6
```

The region above represents all possible pairs of X and Y that result in three pieces shorter than 2 units. The horizontal axis represents the position of X, and the vertical axis represents the position of Y.

The shaded triangle in the region represents the pairs of X and Y that satisfy the three inequalities above. This is a right triangle with base 4 and height 2, so its area is:

(1/2) * 4 * 2 = 4

The total area of the region above is (1/2) * 6 * 6 = 18. So the probability that all three resulting pieces are shorter than 2 units is:

P(X, Y satisfy the inequalities) = Area of the shaded triangle / Total area of the region
                                 = 4 / 18
                                 = 2 / 9

Therefore, the probability that all three resulting pieces are shorter than 2 units is 2/9.

 Mar 16, 2023

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