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Buffon's needle problem

Question

Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?
(Quoted from Wikipedia)

(Source: Wikipedia)

Experiments

You need: handout, 10 sticks of length 10cm per group

1. Pick a bunch of 10 sticks, throw it on the handout.
2. Record the number of stickes that touch any parallel line in the column of "touched". Then write 10 in the column of "needles".
3. Repeat the previous two steps for 10 times.
4. Compute the sum of "touched" and sum of "needles". Then exam the estimated probability.
5. If there are several groups, we may combine the data together.

Intuition

Each stick falling on the paper has two parameter controlling it:

1. The distance between the stick center to the shortest parallel line.
2. The angle of the stick.

Using the distance as the

$x$-axis and the angle as the

$y$-axis, we are able to draw a rectangle–-this is the sample space. For each point on the sample space, identify the point such that the corresponding stick configuration make it touch a parallel line. Then we may use the area to compute the probability.

More questions to think about

1. For each point on the sample space, determine if it corresponding to a stick that touch a parallel line.
2. Describe the set by a function.
3. Use integration to caculate the area and the probability.
4. Now can you explain why an event can still happen even if its probablity is zero?

Resources

1. YouTube: Buffon's needle problem by Jephian Lin