Probability of picking 2 numbers between 0 and 1 to be within 1/2 distance of...

Libby Owens

Libby Owens

Answered

2022-07-18

Probability of picking 2 numbers between 0 and 1 to be within 1/2 distance of each other?
What's the probability of picking 2 numbers, x and y, between 0 and 1 such that they will be within the distance of 12 of each other?
In other words, Pr ( distance between x and y 1 2 ) = ?
I solved the problem through a geometric approach by rewriting the probability as Pr ( | x y | 1 2 ) and graphing | x y | 1 2
From the graph, I calculated the red area to be 75%.
Question: What would be a non-geometric solution to this problem?

Answer & Explanation

iljovskint

iljovskint

Expert

2022-07-19Added 18 answers

Step 1
If you pick two numbers greater than 1 2 , subtract 1 2 from the smaller of the two. If they are the same, subtract 1 2 from the first number. This is a bijection to the two numbers being over 1 2 apart.
Step 2
The probability of this not happening is 1 ( 1 2 ) 2 = 3 4

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