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Geometric Probability

Sometimes, probability questions can be interpreted geometrically, from simple examples like throwing darts to surprising applications like catching the bus!

2-dimensional Geometric Probability


If we randomly choose two numbers \(a\) and \(b\) from the interval \([0,1],\) what is the probability that the quadratic \(x^2+ax+b=0\) has two distinct real roots?

What is the probability that \(x^2+ax+b=0\) has no real roots when \(a\) and \(b\) are two randomly chosen numbers from the interval \([0,2]?\)

Rachel and Ross will go to the gas station at a random time between 4 and 5 o'clock. If it takes exactly ten minutes for each person to refill his/her car and get out of the gas station, what is the probability that they meet at the gas station?

In the figure above, \(\square ABCD\) is a square with side length 1. Find the probability that \(\triangle PAB\) is an obtuse triangle, when \(P\) is a randomly chosen point inside \(\square ABCD.\)

The figure above shows the base (of radius 11 cm) of a cylindrical bucket. There is a red circle of radius 4 cm centered at the center of the base. If Tom throws a coin of radius 1 cm into this bucket, what is the probability that the coin will land in contact with the perimeter of the red circle?


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