1. Attention: We have put together a thread with tips and a tutorial video to help with using the new software. Please take a moment to check out the thread here: Trapshooters.com Tutorial & Help Video.
    Dismiss Notice

Is Trapshooting Binomial?

Discussion in 'Uncategorized Threads' started by Pocatello, Jun 14, 2007.

Thread Status:
Not open for further replies.
  1. Pocatello

    Pocatello Active Member

    Joined:
    Jan 29, 1998
    Messages:
    1,547
    This relates to a proposed mathematical probability model for trapshooting discussed in another thread (above), so if you don't care about that sort of stuff, you can stop reading now.

    If you are still here, suppose we have a 100 target event, singles or handicap, shot in four subevents of twenty-five targets each. If the process is binomial, we could calculate the probability of a shooter breaking any given score once we know his/her probability of breaking any single target. Our best guess for that probability is the long-term proportion of such targets broken, i.e. the "average". For example, my ATA singles average for the past five years or so has hovered around 94, so if we take my probability to be p=0.94, the my probability of breaking 98 out of 100 would be about 0.041 (i.e. I have about a 4.1% chance of breaking a 98). However there are only two different ways a shooter can break 98 out of 100 on four subevents of 25 birds each:

    1) They miss both birds on one field

    2) They miss one bird each on two separate fields

    One can calculate the probabilities of each of those cases happening, if we happen to know the probability p of breaking a single bird, but the interesting thing is that no matter what p is, missing one on two separate fields is about three times as likely as missing two on one field. In particular, no matter the value of p, on all scores of 98, 75.8% of all scores of 98 should have one miss on two separate fields, and 24.2% should have both misses on one field. Now understand that we are talking about a probability distribution here. On any given event things must fall into one or the other category, but over a large number of such events, the distribution will be close to the percentages mentioned above. It's like a Roulette wheel in a casino. On any given spin the outcomes are random, but over a large number of spins they are very predictible indeed, a little less than 47.4% red, 47.4% black, and 5.3% 0 or 00.

    For scores of 97 on four fields, there are only three possibilities:

    1) All three misses on one field, happening 5.7% of the time;

    2) Two misses on one field and one on a second field, about 55.7% of the time;

    3) One miss on each of three separate fields, about 38.7% (all percentages rounded to the nearest tenth).

    For scores of 96, there are five possibilities:

    1) Four misses on one field, about 1.3%;

    2)Three misses on one field and one on a second, about 17.6%;

    3) Two misses on each of two separate fields, about 13.8%;

    4) Two misses on one field, and one miss on each of two others, about 57.4%;

    5) One miss on each field, about 10.0%.

    Similar numbers can be calculated for any other score, and the distribution can be calculated, independent of the probability p. If shooting is binomial, the distribution of given scores for a large tournament will be close to the theoretical. It should be easy to get such data, say the scores from the 2007 Grand, by field. A little number crunching would tell a lot. Anybody interested?
     
  2. revsublime

    revsublime TS Member

    Joined:
    Aug 21, 2006
    Messages:
    1,042
    can anything with "good days" and "bad days" or good luck & bad luck be considered in any way to be binomial?

    Seems like you'd have to have a very staunch and dependable "control" for this to work.
     
  3. halfmile

    halfmile Well-Known Member

    Joined:
    Jan 29, 1998
    Messages:
    15,642
    Location:
    Green Bay Wisconsin
    Someone has WAY too much time on their hands.

    HM
     
  4. phirel

    phirel TS Member

    Joined:
    Jan 29, 1998
    Messages:
    9,556
    Pocatello- Your suggestion would be way too much work for me. I do not like to rely on anecdotal observations, but in this case I will. Unless I am shooting very well in an event, my results are far from binomial. The misses come in clusters, especially in doubles and handicap. Other shooters seem to demonstrate the same problem.

    Pat Ireland
     
  5. Harold

    Harold TS Member

    Joined:
    Jan 29, 1998
    Messages:
    247
    I'd be curious to see the results. Not curious enough to do the work though.

    I once read an article, I think it was in Sports Illustrated, where they tried to see if there was such a thing as a hot hand in basketball. They concluded that the apparent hot or cold was a myth. The scores could be accounted for by just randomness assuming the probability of a basket was the player's shooting average. I'm not sure I agree or not.

    A couple of factors could go against the binomial distribution- the shooting conditions and a player who is improving rapidly. But these should not come into play in a single event.
     
  6. balance365

    balance365 TS Member

    Joined:
    Jan 9, 2006
    Messages:
    70
    Every trial is an independent event, with a "true"/"false" result. By strict definition, that is a binomial event, thus a binomial distribution is correct.


    The problem with this model, for predicting results, is the assumption that the average results of past performance can predict future events. Where this fails is a)target presentations are not the same every time, weather variables affect flight and appearance; b)shooter performance changes over time (some for the better, some for the worse, due to any number of physical or mental "ailments"). Also, as I stated in other posts, this model is further eroded by the distribution of scores, first, there is a skewed distribution, and second, the standard deviation of the population of shooters is pretty wide. With such conditions, the average may not be the best predictor. I haven't done any crunching, but maybe a median socre may be a better predictor, at least until I get a chance to run some numbers, its a hypothesis.


    In your example, you assume that each sub event is shot on a different field, correct? Would shooting a bank system, since you are down to 2 fields, theoretically reducing the variation in presentation of targets, affect the probabilities?
     
  7. fssberson

    fssberson Active Member

    Joined:
    Jan 29, 1998
    Messages:
    2,375
    Now if they could come up with a "binomial" prediction for keeping my head down on the gun, then all of their other figuring would be meaningless. Fred
     
  8. Pocatello

    Pocatello Active Member

    Joined:
    Jan 29, 1998
    Messages:
    1,547
    Pat, in the other thread I stated that my shooting is far from Binomial also, for the same reason. When I miss, I tend to start to miss more.

    Balance365, in the other thread it is the independence assumption I question. And no, there is no assumption that the subevents are shot on different fields. In fact, there is an implicit assumption that many different shooters are shooting them, likely on many different fields, but the p value for a shooter is consistent from field to field. If it is not, there is no validity to treating the whole event is binomial. The point is that if shooting is binomial, then it doesn't matter what shooter is breaking a score, say a 97 for example. In the long run 5.7% of all 97 scores will be three misses on one of the fields, 55.7% will be two misses on one field and one miss on another, and 38.7% will be one miss on three separate fields, NO MATTER WHO IS DOING THE SHOOTING OR WHAT HIS/HER AVERAGE IS (Caps for emphasis - sorry). It would be relatively easy to get data for a whole bunch of 97 scores and see whether they fit the pattern. Now 97 is a reasonably good score, at least for me, and as I said in the earlier thread I believe the better shooters more nearly approach the binomial. I believe the data would show approximately what theory predicts, although my gut feeling is that we would see the 2 1 misses a little higher, and the three on one trap a little lower. However if we look at more typical C D class scores, say around 90, I'd expect the clustered misses would be significantly higher than predicted, and the distributed misses lower.

    Anybody up for looking at some real data?
     
  9. DanR

    DanR TS Member

    Joined:
    Jan 29, 1998
    Messages:
    45
    Pocatello,

    Can you please clarify what you hope to prove or disprove?

    I'm a little puzzled by what you are trying to estimate or test with the "number crunching" which you suggested. If testing is your objective, what is your null hypothesis? It sounds to me that your Ho is: The distribution of lost birds per trap is independent of the probabibility and therefore not binomial, and H1: the distribution is a function of the probability and therefore trapshooting is binomial. By definition, trapshooting seems to be binomial, and consequently over a relatively small number of repititions (e.g. 100 or so) and any sample mean remotely different from 50% (a random distributioin) the sample means of binomial distributions turn out to be statisticially significant. That's why political pollsters can predict elections with such small samples.

    I said that trapshooting "seems to be binomial" because the requirements for a binomial distribution are as jbbor described above, but in trapshooting the probability of breaking a target may not be independent of whether or not the previous target was broken. That is, it might be more likely to miss a target if one has missed the previous one. So, perhaps a test for independence would be a more pertinent question. Unfortunately, I can not for the life of me remember what test would be used to determine that. I do not think that the Chi Square goodness of fit is the right one for that.

    - Dan R.
     
  10. Pocatello

    Pocatello Active Member

    Joined:
    Jan 29, 1998
    Messages:
    1,547
    Gary, it works for me, but that may be just because I accessed it through the "Edit/Delete your messages" feature. The thread title is "New Thought on 16 yd Classification", and my reply to it posted June 13 at 11:47 AM MT.
     
  11. Pocatello

    Pocatello Active Member

    Joined:
    Jan 29, 1998
    Messages:
    1,547
    Dan, as stated above, it is the independence assumption I doubt, for just the reason you state, and my curiosity is purely intellectual. If the process is binomial, then the probability distribution of lost birds over four fields IS independent of p (mathematically provable - I can send it to you if you are interested), and in a large enough sample the sample distribution will reflect the probability distribution.
     
  12. balance365

    balance365 TS Member

    Joined:
    Jan 9, 2006
    Messages:
    70
    Thanks Pocatello. I see what you are saying, and I think I have some small shoot (about 80-100 persons) results I can play with, just may take a couple of days to do (OK its not my first priority, but I am interested). I also would like to play with the idea of median score as p instead of average score. Can some people e-mail some detailed shooter records to see if that is a more accurate predictor of outcomes than average. I would only be looking at singles socres.
     
Thread Status:
Not open for further replies.