The probability that all of the resulting pieces have lengths of at least 8.5 cm. is 0.0625
We can think of the probabilities as direct areas because the distribution is uniform (as it is along a straight line).
A square with an area of 34² can be used to represent it.
Total stick area = length x width = 34 x 34 = 1156 cm².
This area has a probability of 1.
When two points are chosen at random, they can now be represented as X₁ and X₂.
There will be three pieces of the stick after breaking from two points.
The length of 1 piece = 8.5 cm
Then the length of 3 pieces = 8.5 x 3 = 25.5 cm
Now the area of pieces with a minimum length of 8.5 cm
= (length of whole stick - length of 3 pieces)² cm²
= (34 - 25.5)² cm²
= 8.5² cm²
= 72.25 cm²
Probability of obtaining pieces of at least 8.5 lengths is
P(8.5 lengths)
= Area of pieces with 8.5 lengths/Area of stick
= 72.25/1156
= 0.0625
Therefore the probability that all of the resulting pieces have lengths of at least 8.5 cm. is 0.0625
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