"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?"
The solution for the sought probability $p$, in the case where the needle length $l$ is not greater than the width $t$ of the strips, is $\frac{2}{\pi}\cdot \frac{l}{t}$
So far so good.
What I don't understand:
Let's go back to the basics:
The probability of an event $E = P(E)$
$$P(E) = \frac{n}{N}\\
n = \text{Number of outcomes that count as event }E\\
N = \text{Total number of outcomes.}$$
An example:
What is the probability of drawing a white ball from a bag that contains $3$ white balls and $2$ black balls?
Probability of drawing a white ball $= P(W)$, with $n = 3, N = 5$.
$$P(W) = \frac{n}{N} = \frac{3}{5}$$
Notice $P(W)$ is (a) rational (number) which indicates outcomes can only be positive integers and so (?) probabilities can only be rational numbers.
How come there's a noninteger number of outcomes in the solution to Buffon's needle (there's a $\pi$ in the denominator, which I understand to mean that if $t = 1$ then the sample space (total number of possible outcomes) is $3.14159\dots$??? How? 🤔
Best Answer
When the number $N$ of outcomes is finite (so the number $n$ of outcomes that "lead to E" is also finite) and all outcomes are equally likely, then you can use the formula
$P(E) = n / N$.
However, in more general case (either with $N$ being infinite and/or having different probabilities for each "small event"), the formula above should be read as
$P(E) = $ mass of outcomes that lead to E / total mass of outcomes.
By definition, the total mass of outcomes, i.e., probability of the event that always happens, is 1, so the formula is actually
$P(E) = $ mass of outcomes that lead to E.
How exactly is this "mass" computed, is another question. See Wikipedia page for Probability distribution and dive into