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

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?

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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?

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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 \(\square ABCD.\)

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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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