Exercising my mind a little bit with the problem and I've come to a dilemma. Picture the following scenario:
We have a standard dart board. We simulate the dart board and throw 100,000 darts at it, using predetermined random distributions for the x- and y-axis. Every dart lands on or around the board, garnering an amount of points (0, 1-20 times a 1-3 multiplier, 25, or 50). Throws are IID events.
The question is: what is the probability of hitting a certain spot, or certain 3 spots (given three darts in reality)?
What I've gotten so far is the simulated dart throws and for each dart board space, the number of darts which landed on that spot. For example, of 100,000 throws: 10,429 landed outside the board, 3358 landed on a single multiplier of 20, 120 landed on a 20 double multiplier, and 344 landed on a 20 triple multiplier.
Given these numbers, I can confidently say the probability of hitting a triple 20 is $\frac{344}{100000}$. However, I can't decide the proper way of determining the probability of hitting say, two triple twenties and a double twenty.
My gut feeling is this is the intersection of events, thus
$$
P(\text{triple 20} \cap \text{triple 20} \cap \text{double 20})=(\frac{344}{100000})^2(\frac{120}{100000})
$$
I would take this as it is it, however this is approximately $1.42\times 10^{-8}$. I have a hard time believing this from my own dart playing experience. I have not thrown over 10 million darts and yet I've landed things like this before.
From my doubt, I thought: maybe this is a union of events:
$$
P(\text{triple 20} \cup \text{triple 20} \cup \text{double 20})=(\frac{344}{100000}) + (\frac{344}{100000}) + (\frac{120}{100000})
$$
which is a much more believable 0.27% probability of occurring.
What am I overlooking? What haven't I considered? Why does my math seem so uncertain and why is/are one or both of these incorrect?
Best Answer
Assuming that the results of each dart throw are independent*, then the probability of hitting double-double-treble 20, in that order, is indeed simply the product of the individual probabilities that you’ve derived from your simulation. I would argue that the resulting very small value does in fact match your experience: you’ve computed the probability of hitting that particular combination while aiming for the bullseye. The relevant real-life experience to which you should compare this isn’t the overall frequency with which you can land this particular combination, but instead the number of times that you’ve gotten it accidentally while trying to shoot bulls. I’ll go out on a limb here and say that’s never happened. The first two darts both have to land in a fairly small area that’s half a board width away from your aiming point.
Note, too, that your simulated probability of a treble 20 is almost three times that of a double 20. This certainly makes sense for Gaussian scatter from the center of the board, but is backwards from what one might expect when actually trying to hit those regions. Cetera paribus the probability of a treble 20 should be smaller since it covers a significantly smaller area than the double 20.
* Assuming independence for such small regions of the dart board doesn’t seem like a good approximation to me. Each dart that lands significantly reduces in interesting ways the available area for the next one. There’s not a lot of leeway for three darts in any of the treble zones, for instance.