# Inside a square of side 2 units, five points are marked at random. What is the probability that there are at least two points such that the distance between them is at most sqrt{2} units?

Geometric probability
Inside a square of side 2 units , five points are marked at random. What is the probability that there are at least two points such that the distance between them is at most $\sqrt{2}$ units?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Matthias Calhoun
Step 1
$p=0$ because the furthest separation between 5 points in a square is with one at each corner and one in the middle, but by Pythagoras theorem the distance between the corner and the middle is $\sqrt{2}$.
Step 2
Not a formal proof I grant you but the logic is infallible.
###### Did you like this example?
Kelton Molina
Step 1
Divide the square into 4 squares of side length 1.In at least one square there are two or more points.
Step 2
All points in the same square have distance less than ${2}^{0.5}$. So the answer is 1