# Probability of darts hitting a certain spot (or several spots)

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?

• Do the distributions that you’re working with reflect the fact that you’re aiming for those particular regions?
– amd
Commented Jun 16, 2018 at 19:50
• @amd I'm using Generalized Normal distributions with means centered at the bullseye and beta and SD tuned to roughly fit what I expect out of my own games (some darts completely missing the board, a majority of them getting somewhere on the board). Commented Jun 16, 2018 at 19:55
• So, you’re effectively aiming at the center of the board. Do you often hit the double and treble rings when you do that in real life? What I’m trying to convey is that your experienced-based intuition of the probability of trip-trip-dub is likely based on a very different set of experiments.
– amd
Commented Jun 16, 2018 at 20:04
• I'll admit it's a simplified model, and in reality I probably hit the multipliers more (my simulation hits triples 7.6% of the time, and doubles 3.8% of the time). I tracked about 150 darts in reality to see where I was hitting and the distributions I'm using aren't too far off (although I do get fewer bullseyes than the model would claim). There's quite a bit of randomness in my own throws though. Commented Jun 16, 2018 at 20:10
• To put it in concrete terms, $344/100000$ is the probability of hitting a triple 20 when aiming at the bullseye. My intuition is that this should be quite small, and the probability of hitting a double 20 even smaller, unless you’re really bad at darts (or very drunk).
– amd
Commented Jun 16, 2018 at 20:11

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.

• "the product of the individual probabilities"... this is the affirmation that I was looking for, however could you add a bit about why (and why it's not a union)? Seems I'll have to record throws in reality more to get a better sense of the true distribution. Commented Jun 18, 2018 at 15:51
• @AlexL This is simply an application of the multiplication rule for independent probabilities. As to why intersection is appropriate, the general rule of thumb is that “and” translates to intersection of events, while “or” translates to union of events.
– amd
Commented Jun 20, 2018 at 1:38
• @AlexL I think you can get a reasonable model by using the suggestion that I made elsewhere: center the distribution on the aiming point. This means that when you shift your aim you’ll need to recompute probabilities, but that’s OK. From my own experience, I’d bias it a bit to make hits below the aiming point more likely than above, but that’s secondary fine-tuning.
– amd
Commented Jun 20, 2018 at 1:43