Find the probability of distance of two points, which are selected in [0,a] closed interval, is less than ka k<1

clovnerie0q 2022-10-08 Answered
A geometric probability question
Find the probability of distance of two points ,which are selected in [0,a] closed interval, is less than ka k < 1.
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Answers (2)

Elliott Rollins
Answered 2022-10-09 Author has 8 answers
Step 1

The region of | y x | k a looks similar to the picture that I have attached. We just have to use the total area to subtract away the area of the area of the two triangles.
Step 2
a 2 2 1 2 ( a k a ) 2 a 2 = 1 ( 1 k ) 2 = k ( 2 k )
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solvarmedw
Answered 2022-10-10 Author has 3 answers
Step 1
We need to find the measure of A, which is given by A d x d y = 0 a max ( y k a , 0 ) m i n ( y + k a , a ) d x d y = 0 a min ( y + k a , a ) max ( y k a , 0 ) d y . .
Step 2
Note that 0 a min ( y + k a , a ) d y = 0 a k a y + k a d y + a k a a a d y .
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