# Modeling distribution on a dart board

I have statistics for a game similar to lawn dart (3 concentric rings instead of 2) and I wanted to simulate shot distribution. What statistical distribution could I use?

The game is throwing a magnet at a mini fridge with 3 concentric rings. We have 3 players that play this game and I have been collecting data on it for fun. If the outer ring has radius 1 then middle ring has radius of 0.368 and inner has radius of 0.105. Hitting a ring is worth 10, 25 and 50 points respectively. At this point I record the point outcome of the game and each individual shot on the board (which point value) which gives me shot percentage. Outer ring is hit 57.35%, middle ring is hit 7.84% and inner is hit 0.98%.

What I want to do with this is make a model using some statistical distribution that will generate an (x, y) coordinate based on player performance. Then simulate a few round robin style tournaments, etc... and in general predict outcomes and see if the real stats match to the simulated stats. The issue is that using a Gaussian distribution doesn't give an accurate enough estimation of the performance of the players. It can approximately fit 2 of the 3 statistical points but does not seem to fully characterize the distribution. Is there a better curve/method of fitting this data?

• What data do you have exactly? – David Jul 3 '19 at 14:38
• What exactly are the values of these "statistical points" and how were they measured? What are the dimensions of the target? – whuber Jul 3 '19 at 14:38
• If the outer ring has radius 1 then middle ring has radius of 0.368 and inner has radius of 0.105. Outer ring is hit 57.35%, middle ring is hit 7.84% and inner is hit 0.98%. Percents are just #hit in particular ring / #of shots. – Boto Jul 3 '19 at 15:28
• It seems to me you have clearly described one of the best possible models: it remains only to compute the chance of missing the board altogether, equal to $(100 - 53.75-7.84-0.98)\%=37.43\%.$ There's nothing else one can add on the basis of the information you have collected. This is a discrete distribution with four outcomes having the four given probabilities. For simulations, see stats.stackexchange.com/questions/26858 and stats.stackexchange.com/questions/67911. – whuber Jul 3 '19 at 19:07
• Alright. I'll do that first. But, I am still curious if there is a continuous distribution that could fit the data by generating 2d coordinates. In my mind that gives it more flexibility if I was to mess around with variations of the game such as ring size. – Boto Jul 3 '19 at 19:23

According to Schwartz 1, the Rayleigh distribution has been used to model the distribution of bullets hitting a target. The model assumes the bullet hits are gaussian distributed along the target’s horizontal axis. Likewise and independently, the bullet hits are gaussian distributed along the target’s vertical axis. The two distributions are centered on the origin (center) of the target and their mutual population standard deviation is $$\sigma$$. For darts hitting a dart board, the model is the same. Obviously, gravity would cause a downward bias, but it is either neglected in the model or perhaps is considered to be a correctable issue. Either way, it is not considered further.

The Rayleigh distribution is given by 1 $$p_r(r) = \frac{r}{\sigma^2} e^{-r^2 / 2\sigma^2}$$

Now consider the following figure, which depicts the target and two circles, with radii $$r_1$$ and $$r_2$$, centered on the origin.

Defining $$A_1$$ as the area inside the smaller circle, the area outside the smaller circle is $$1 - A_1$$, which is given by the following integral [2]:

$$1 - A_1 = \int\limits_{r_1}^{\infty } p_r(r) \mathrm dr = e^{-r_1^2 / 2\sigma^2}$$

Hence, $$A_1 = 1 - e^{-r_1^2 / 2\sigma^2}$$. Defining $$A_2$$ as the area inside the larger circle, it follows that $$A_2 = 1 - e^{-r_2^2 / 2\sigma^2}$$, so the area of the annulus, between $$r_1$$ and $$r_2$$, is $$e^{-r_1^2 / 2\sigma^2} - e^{-r_2^2 / 2\sigma^2}$$. Hope this helps!

References:

1. M. Schwartz, Information Transmission, Modulation, and Noise, McGraw-Hill Book Company, New York, 3rd Ed., ©1980, p. 379.
2. A.D. Whalen, Detection of Signals in Noise, Academic Press, New York, ©1971, p. 204.
• A sceptic might announce that it's unlikely these data would be accurately modeled with this one-parameter distribution. What rejoinder would you offer? That is, how would you propose determining whether your answer is reasonable? – whuber Jul 5 '19 at 14:03
• @whuber I wouldn’t, actually! I think it is a pretty bad model, unless darts were dropped vertically onto a dart board that was flat on level ground and, in addition, random deflections were demonstrably as stated. Not much chance of that! I was thinking about this as I wrote my answer, but thought that even a poor model might spark a good question down the road. – Ed V Jul 5 '19 at 14:09