The problem in more mathematical terms is: Given a needle of length dropped on a plane ruled with parallel lines t units apart, what is the probability that the needle will cross a line?
Let x be the distance from the center of the needle to the closest line, let θ be the acute angle between the needle and the lines, and let .
The probability density function of x between 0 and t /2 is
The probability density function of θ between 0 and π/2 is
The two random variables, x and θ, are independent, so the joint probability density function is the product
The needle crosses a line if
Integrating the joint probability density function gives the probability that the needle will cross a line:
For n needles dropped with h of the needles crossing lines, the probability is
which can be solved for π to get
Now suppose . In this case, integrating the joint probability density function, we obtain:
Thus, performing the above integration, we see that, when , the probability that the needle will cross a line is
Mario Lazzarini, an Italian mathematician, performed the Buffon's needle experiment in 1901. Tossing a needle 3408 times, he attained the well-known estimate 355/113 for π, which is a very accurate value, differing from π by no more than 3×10−7. This is an impressive result, but is something of a cheat, as follows.
Lazzarini chose needles whose length was 5/6 of the width of the strips of wood. In this case, the probability that the needles will cross the lines is 5/3π. Thus if one were to drop n needles and get x crossings, one would estimate π as
π is very nearly 355/113; in fact, there is no better rational approximation with fewer than 5 digits in the numerator and denominator. So if one had n and x such that:
one would derive an unexpectedly accurate approximation to π, simply because the fraction 355/113 happens to be so close to the correct value. But this is easily arranged. To do this, one should pick n as a multiple of 213, because then 113n/213 is an integer; one then drops n needles, and hopes for exactly x = 113n/213 successes.
If one drops 213 needles and happens to get 113 successes, then one can triumphantly report an estimate of π accurate to six decimal places. If not, one can just do 213 more trials and hope for a total of 226 successes; if not, just repeat as necessary. Lazzarini performed 3408 = 213 · 16 trials, making it seem likely that this is the strategy he used to obtain his "estimate".