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The Mathematics of Missing Targets.

Discussion in 'Shooting Related Threads' started by JBrooks, May 3, 2009.

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  1. JBrooks

    JBrooks TS Member

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    Regardless of what your singles average is, I think most experienced shooters would agree that hitting 90 out of 100 targets is not very hard. In fact, better shooters would consider hitting 95 out of 100 targets to be a reasonably expected outcome on a fair weather day and the best shooters seldom expect to hit less than 98 targets.

    However, why is hitting those remaining 10, 5 or 2 so difficult? Is is just the law of large numbers or do we actually psyche ourselves out at some point during the 100?

    If you carry a 98 average and post a 100 should you expect to post another 100 or are you doomed to post a 96? After all, you have a 98 average for a reason.

    Let's say our 98 shooter shoots a 100 on Sunday. He runs another 50 in his Tuesday night trap league. He hit another 50 in practice on Friday and he runs his practice trap on Saturday morning. He has now hit 225 straight yet his average says he should have missed 5 targets. He goes to the line for a 100 target event. Does he actually have much of chance to run another 100?
     
  2. otnot

    otnot Active Member

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    Why is it then that I average 96 but once in a while I can shoot a 100 straight? Or even shoot a 200 straight? Do you only find the zone a couple of times a year or is it just luck? LOL I think it's just the game of trap.
     
  3. Old Cowboy

    Old Cowboy Active Member

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    It's that little rascal sitting up on your shoulder telling you to miss
     
  4. Neil Winston

    Neil Winston Well-Known Member

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    If you want to assume that shots are basically independent, your 98 averager has but about a one in seven chance of 100 today.

    <a href="http://photobucket.com" target="_blank">[​IMG]</a>

    Neil
     
  5. JBrooks

    JBrooks TS Member

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    Interesting chart Neil. But, how is it that those D, C & B shooters put up 98s, 99s and 100s against those odds?
     
  6. Neil Winston

    Neil Winston Well-Known Member

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    Well, for one thing, at least for the lower classes (most of all) the average probably is artificially lowered by some exceptionally low scores. By the time you get to A, though, things begin to fall into place better, at least in the ratio of scores. A person with a 96 will shoot several times as many 99's as 100's, generally.

    As I said, it only works if the shots are independent. If they aren't; there's no answer to your question; "Does he actually have much of chance to run another 100?"

    Neil
     
  7. ccw1911

    ccw1911 Member

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    Mental exhaustion, moving out of your comfort zone, trying to duplicate performance rather than taking each shot as an individual, falling into mindless repitition, more but I can't think of them right now.

    Oh and then there is an old Chinese proverb I will paraphrase. "When on a foot journey of 100 miles it is wise to note that after traveling 99 you are but half way there"
     
  8. JBrooks

    JBrooks TS Member

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    So does the guy with the 98 average have a 98% chance of hitting the next target thrown?
     
  9. PAR8HED

    PAR8HED Member

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    Interesting. On the flip side of that questions is this, why do you improve as the round goes on? Easy answer for me is the nerves settle down and I get into my routine and the round. But it does leave one frustrated when you finish the last 15 and miss 5 of the first 10.
    Hal
     
  10. ccw1911

    ccw1911 Member

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    We tend to focus on our best shooting, there is a lot more ground to gain on the other side, our worst shooting. In other words work on the worst part of your game rather than trying to get that last 2 percent out of the best of your game. Not how can I get those other 2 birds I needed for the 100 today, but how can I improve those 85's I shoot sometimes, find that answer an the other is fixed. JMHO
     
  11. phirel

    phirel TS Member

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    JBrooks- Your first question (shooter with a 98 average shoots 100- can he reasonably expect to shoot another 100 ?). Probability involves future events, not what has happened in the past. The chance that a tossed coin will come up heads is 50%. The probability that you will toss three heads in a row is 1/8. But if you toss three heads in a row, the probability that the next toss will be heads is still 1/2.

    If a shooter has a 98% average, you could conclude with 95% confidence that his next score will be between 98 +- one standard deviation around his average. Raw averages are poor predictors of future events and not good descriptions of past performances. The raw average +- standard deviation, or even better +- standard error is much better.

    Then there are shooters, like me, who defy the laws of probability. Both my bowel movements and my shooting are irregular and often one of them resembles the other.

    Pat Ireland
     
  12. hmb

    hmb Well-Known Member

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    Pat,

    Please try to not discuss your bowel movements on TS.com. Thank you. HMB
     
  13. Pocatello

    Pocatello Active Member

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    Place mark for a later response to this thread.
     
  14. Carol Lister

    Carol Lister TS Member

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    Frankly, I am firmly convinced that more targets are missed because of over-analyzing WHY targets are missed than for any other reason. Why waste heartbeats by over-complicating simply things??

    Carol Lister
     
  15. shot410ga

    shot410ga Well-Known Member

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    For 30+ years I've kept track of my scores (all scores). And the only thing I've been able to determine is a trend line. Down in the winter, up in the summer. I've also looked at trends based on gun modifications or shells, loads and chokes. As and example, if I've changed something on the gun, a three to four month down trend will occure before an up trend starts to take place or I go back to what I had in the first place. I've tried forecasting based on long trend lines, and it doesen't work, for me anyway.
     
  16. Neil Winston

    Neil Winston Well-Known Member

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    JBrooks, you asked "So does the guy with the 98 average have a 98% chance of hitting the next target thrown?":

    . . . and I think the answer is that his chances are at least a bit - and maybe a good bit - higher. An average is reduced below "what it should be" which, for a moment accepting independence of shots, is exactly that, the number which would accurately give the probability of hitting the next target.

    (At least) two things press an average down:

    1. There's a lot more down than up for most averages. One day of bad performance can overpower, average-wise, a lot of good shooting

    2. There's a ceiling effect by which a hundred achieved cuts off chances to see how many more would have been hit.

    But there are some generalities:

    A. Except at the Grand, you can take the 100's and divide by 6 to get the likely number of 200's.

    B. Though a 96 average might seem to predict a lot of 100's, it doesn't.

    The value of the chart is not to put a particular number to any single situation, but rather to see what a huge effect a high average has on the probability of a good score. To me it's a very rough way to guess how much better one shooter is than another.

    And it's what's wrong with handicap but that hardly needs explaining.

    Neil
     
  17. Pocatello

    Pocatello Active Member

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    The mathematical model that generates Neil's table is known as the Binomial distribution. In the Binomial distribution one has an "experiment" with two outcomes, "success" and "failure". The probability of success in any given experiment is a number p, 0 < p < 1. We conduct n independent experiments and ask what is the probability of having k successes in those n experiments, where 0<=k<=n. "Independent" means that the outcome of any one experiment has no influence on the outcome of any other experiment. Shooting clay targets clearly has some of these features. The "experiment" is shooting at the target. "Success" is when we break it, "failure" is when we do not (two outcomes). We do this experiment repeated times - n is 100 in a typical handicap event. The two main questions are:

    1) Is there a fixed probability p of success on any one trial? A prime candidate would be our "average", actually the historical proportion of those type of targets we have broken. For example, last year in handicap I broke 87.68% of 2500 ATA handicap targets, so should expect that for me p in handicap would be somewhere around 0.8768.

    2) Are the trials independent?

    The second question really is the critical one. All the instructors I've taken clinics from, and all the great shooters I've talked to, emphasize we should strive for that goal. "Any target lost is lost forever, so forget about it. Approach every target the same way, focusing on the fundamentals. Don't think about how well or poorly you are doing, put all of you efforts into this one target", and similar advice we've all heard. I'm sure the great shooters do a better job than most of us in living up to that goal, focusing on the present. I know that when I'm doing well, I know it, and start to count targets. I know that when I'm doing poorly, I know it, and start to count misses. There have been very few times when I've been in that "in the moment" mode. My conjecture is that the great shooters behave very much in a binomial mode, and most of the rest of us do not.

    However there is an easy mathematical way of verifying this, and I'm saving it for a project to do after I retire. If shooting is binomial, then for a given score there will be a very definite distribution of scores across subevents independent of the probability p. For example, suppose we shoot at 100 targets in four subevents of 25 each. Suppose we break a score of 96. There are only five different ways we could have done that:

    1) Miss all four on one trap, and run the three others;

    2) Miss three on one trap, one on another trap, and run two others;

    3) Miss two on one trap, two on another trap, and run two traps;

    4) Miss two on one trap, one on each of two others, and run one trap;

    5) Miss one on each trap.

    If shooting is binomial, then the probability of each of those possibilities is independent of the p involved. In particular case #1 should, over the long run, happen about 1.3% of the time, case #2 should happen about 17.6% of the time, case #3 should happen about 13.8% of the time, case #4 should happen about 57.4% of the time, and case #5 should happen about 10.0% of the time. So to check whether shooting is binomial, go to any large shoot and look at all the singles and handicap scores of 96 shot over four subevents of 25 targets each, and see whether they match the given distribution. Similar distributions correspond to other scores. For example, 55.7% of all 97s should be 2 misses on one field, one miss on another field, and two fields straight, 38.7% should be three 24s and one 25, and 5.7% should be one 22 and three 25s (percentages rounded to the nearest tenth).

    For Pat Ireland: There are two approaches to the coin flipping experiment you cite. Probabilty is the one you use. Statistics would ask the question "Given that the coin has come up heads three times in a row, what is the probability it is a fair coin?"
     
  18. phirel

    phirel TS Member

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    Pocatello- If all trap scores are truly binominal, would not the number of scores of 100 be the same as the number of scores of 0?

    And, on the fair coin question, all I can answer is that "In God We Trust".

    Pat Ireland
     
  19. Setterman

    Setterman Well-Known Member

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    How well you see the targets at the location of the shoot, wind, your squad mates, how you physically feel that day, and temperature are all variables that affect score. Temperature may require more clothes and the muscles may be tight.
    For some of us, the "hangover factor" may affect your score too.
     
  20. Pocatello

    Pocatello Active Member

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    Certainly not, Pat, unless p=0.5. If p > 0.5 then the expected number of 100s exceeds the expected number of zeroes, significantly so as p approaches 1.
     
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