It may be objected that it would be absurd to assume the skill of the personnel in one fleet as twice as great as that of the personnel in the other fleet, but it can easily be shown that even so great a disproportion is not impossible, provided the skill in one fleet is very great. The value of superior skill naturally becomes important where the difficulties are great. A very simple illustration is in firing a gun; for even if the skill of one marksman be greater than that of another, it will be unimportant, if the target is so large and so close that even the inferior marksman can hit it at each shot. The probability of hitting a target--so far as overs and shorts are concerned (or deviations to the left and right)--varies with the fraction _a/y_, where _a_ is the half height (or width) of the target, and _y_ is the mean error. The greater the size of the target, and the less the mean error, the greater the probability of hitting. The size of the two targets being fixed, therefore, the smaller the mean error the greater the probability of hitting. The probability of hitting, however (as can be seen by the formula), does not increase greatly with the decrease of error, except in cases where _a/y_ is small, where the mean error is large relatively to the width or height of the target. For instance, if _a/y_ is .1 in one case, and .2 in another case, the probability is practically double in the second case; whereas, if _a/y_ is 1 in one case, and 2 in another, the probability increases only 55 per cent; while if it is 2 in one case and 4 in the other, the probability of hitting increases only 12 per cent. This means that if two antagonists engage, the more skilful should, and doubtless will, engage under difficult conditions, where _y_ is considerable relatively to _a_; for instance, at long range. Suppose that he engages at such a range that he can make 10 per cent of hits--that is, make 90 per cent of misses; and that his misses relatively to the enemy's is as 90 to 95--so that the enemy makes 95 per cent of misses. This does not seem to be (in fact it is not) an extreme case: and yet _A_ will hit _B_ twice as often as _B_ will hit _A_. In other words, the effective skill of _A_ will be twice that of _B_. This illustrates the effect of training--because all that training in handling any instrument can do is to attain as closely as possible to the maximum output of the instrument; and as the maximum output is attained only when the instrument is handled exactly as it should be handled, and as every departure is therefore an error in handling, we see that the effect of training is merely to diminish errors.