SUMMARY .sx The authors discuss the testing of explosives with special reference to the ability of a test to indicate the presence of significant differences in ignition probability and also to the reliability of the test .sx It is suggested that tests requiring low ignition rates , and particularly no-ignition tests , are , as a class , poor discriminators .sx The ability to discriminate can be increased by increasing the number of ignitions accepted as the pass level .sx It is suggested that a test of 26 shots , in which 13 ignitions are permitted , represents a good compromise , in view of the need to keep the number of shots within reasonable limits .sx 1 .sx INTRODUCTION .sx About a hundred million shots a year are fired in British mines and usually about 6 ignitions are reported each year .sx It is clear that with a practical ignition rate of roughly 10 :sx -7: , a test no more severe than practical use required an impossibly high number of shots to give a reliable answer ; and therefore the test must be made so much more severe ( i.e. the ignition rate in the test must be made so much higher ) that an effective assessment of the safety of an explosive may be made with a practicable number of shots .sx In rigorous terms this thesis demands that the ignition rate be multiplied ten million times or so .sx The multiplying factor can be made up by ( =1 ) Ensuring the presence of practical conditions which are dangerous but rare , e.g. the presence of considerable volumes of an explosive mixture of methane/ air , the absence of stemming in the shothole , and so on .sx ( =2 ) Modifying the test apparatus to increase the ignition rate , e.g. firing the shot in a steel cannon instead of the rock or coal in which it is fired in the mine .sx All of these devices are used in explosives testing ; but apart from some tentative results recorded in the literature ( Cybulski , 1959 ; Schultze-Rhonhof and others , 1959 ) no firm estimate can be made of the relative contributions they make to the multiplying factor .sx However it is probably wise to assume that the contribution of the second group is substantial rather than preponderant .sx This is fortunate rather than the reverse because scientifically any process that extrapolates a million times may be expected to require a lot of proving .sx British approval tests have been such that an explosive is failed if ignitions are obtained in any of the tests .sx This reliance on no-ignition tests has been an almost uniform feature of explosive testing throughout the world although the French system permits ignitions in one of the tests , and recently the United States Bureau of Mines has made a decided break with tradition in this regard ( United States Bureau of Mines , 1961) .sx For the past three years a detailed study of the testing procedure has been conducted at S.M.R.E. ; particular attention has been paid to the statistical problems raised by no-ignition tests .sx It has been concluded that the no-ignition test , as applied to explosives , gives too little information about the ignition probability of the material tested , and that this weakness cannot be removed by any practicable increase in the number of shots fired .sx 2 .sx RELIABILITY AND DISCRIMINATION .sx A good test should meet , 6inter alia , two requirements :sx ( =1 ) It should be reliable , i.e. a repeat test of the same material should give the same result .sx ( =2 ) It should have adequate discrimination , i.e. it should indicate the presence of significant differences .sx No measurement is exactly reproducible , since all are subject to random errors .sx In explosive testing random error appears as a variation in the number of ignitions obtained in repeated tests on identical material .sx However often a trial is repeated , one can never say how many ignitions will take place ; but , at the same time , the more often a trial is repeated , the more exactly can the probability of ignition by an individual shot be stated .sx Once this probability of ignition by an individual shot is known it becomes possible to calculate the probability of any particular number of ignitions in a given number of shots .sx Alternatively , it is possible to calculate the number of shots that must be fired to achieve a given probability of a particular number of ignitions .sx In this situation , complete reliability of acceptance or rejection is impossible ; one may assign only the probability with which material of specified characteristics shall be accepted or rejected .sx This probability can , by firing enough shots , be made to approach certainty as closely as is desired , although a situation is rapidly reached where an enormous number of shots must be fired to achieve a small improvement .sx It is also fundamental that the acceptance and rejection limits cannot be equal although , again , by firing enough shots they may be made to approach each other as closely as is desired .sx The difference between the acceptance and rejection levels is analogous to discrimination .sx Whatever values of ignition probability are chosen as the rejection and acceptance limits and whatever level of probability be chosen for the rejection or acceptance at those limits , material with an ignition probability equal to the mean of the limits will be almost as likely to fail as it is to pass .sx This again is fundamental to all systems of assessment .sx It will be seen therefore that the concepts of reliability and discrimination as applied to testing are complex ones :sx overall , a system can be made reliable to a chosen extent at the limits of a chosen range .sx 3 .sx EXAMINATION OF THE NO-IGNITION TEST .sx In the last section it was pointed out that the reliability of rejection or acceptance is a matter of choice , and clearly opinions will differ as to the desirable level .sx However , it appeared reasonable to the present writers to require that the test should have a 0.95 probability of rejecting an explosive having an ignition probability at the chosen reject level .sx Correspondingly there should be a 0.95 probability of accepting an explosive at the acceptance level .sx Calculations were then made which permitted the plotting of Curve 1 in Fig. 1 .sx In this figure the true probability of ignition with a single shot is plotted against the number of shots of the explosive that must be fired to give a 0.95 probability of one or more ignitions .sx For example a " no-ignition " test of 28 shots will reject , 19 times out of 20 , an explosive with an ignition probability of 0.1 ( for the rest of this paper 19 times out of 20 will be called " reliable " rejection or acceptance .sx ) Curve 2 in Fig. 1 shows the number of shots for which the probability of one or more ignitions is 0.05 , i.e. there is a probability of 0.95 of acceptance .sx From these curves it will be seen that although a 28-shot sequence will reliably reject an explosive of ignition probability of 0.1 , it will not reliably accept explosives until the ignition probability has fallen to 0.0018 ; in other words , if a manufacturer submits an explosive that has a slightly lower ignition probability than 0.1 , he has a moderate chance of getting it through the test but if he submits another that is ten times better in this respect , he has a fair chance of having it rejected .sx Summarizing , if the probability is lower than 0.0018 or higher than 0.1 , the explosive will be reliably passed or failed , but if it has an intermediate value , the test will not give reliable results .sx The curves in Fig. 1 also show that the rejection level and the number of shots in the test may be varied over a wide range but without an appreciable change in the value of approximately 50 for the ratio of the acceptance to the pass level .sx It appears to be impossible to avoid poor discrimination with no-ignition tests .sx 4 .sx TESTS PERMITTING IGNITIONS .sx In the last section it was found that poor discrimination appeared to be a characteristic of no-ignition tests :sx the effect of permitting one ignition is shown in Fig. 2 and Fig. 3 shows the characteristics for 2-ignition tests .sx It will be noted that the gap between the rejection and the acceptance curves narrows , i.e. the discrimination is improved when the number of permitted ignitions is increased .sx The calculations on which Fig. 2 and 3 are based have been extended , and the results are summarized in Table 1 .sx The accuracy of discrimination steadily increases with the number of ignitions ( m ) accepted as the pass level .sx Confining attention for the time being to a reliable rejection level of p;r ; equal to 0.1 , Table 1 shows that the ratio ( p;r;/ p;a; ) does not fall to the neighbourhood of 2 until the number ( n ) of shots fired is nearly 200 and the acceptable number ( m ) of ignitions rises to 12 .sx The table does not extend beyond the point where ( p;r;/ p;a; ) falls to the neighbourhood of two because this seemed a good compromise , as far as explosives are concerned , between the requirements of discrimination and the need to keep the number of shots within practicable limits ; in view of the variabilities inherent in the conditions of use , perhaps it should not be taken too seriously if the value of ( p;r;/ p;a; ) for a given explosive fluctuates in the range of 2 to 1 .sx The following example may illustrate the operation of a test with a pass level of not more than 12 ignitions in 200 shots .sx This test has a reliable p;r ; of 0.1 and a reliable p;a ; ( acceptance level ) of approx 0.05 ; for reliable acceptance the manufacturers must work to an ignition probability per shot ( p ) of 0.05. If the product deteriorates , and is then re-tested , there is a probability of 0.95 that the deterioration will be detected when the ignition probability has increased by a factor of 2.0. To a considerable extent the sensitivity of existing explosives tests is adjustable at will , usually by adjusting the charge weight but also by changes in the test apparatus .sx What are the consequences of changing the sensitivity ?sx Table 1 gives the appropriate figures for rejection ignition probability of 0.5 and shows that equally good discrimination can be obtained but with far fewer shots .sx Table 1 indicates that an economical and discriminating test at a rejection level of p;r ; = 0.5 is to fire 35 shots and permit 12 ignitions .sx The calculations have since been extended by Mr. G. Fogg of S.M.R.E. and it appears that at a rejection level of p;r ; = 0.673 a discrimination ratio of 2 is obtained with a round ( n ) of 26 shots and a permitted number ( m ) of 13 ignitions .sx 5 .sx MATHEMATICAL BASIS .sx The mathematical basis on which Figs 1 , 2 and 3 and Table 1 were calculated is simple and well-known ; see for example David , F.N. ( 1949) .sx The probability , P , of an explosive being accepted after a series of tests is a calculable function of the probability of ignition in a single test , p , and of the standards required in the series .sx For example , if our standard requirement is 0 ignitions in n trials , we have For sufficiently large p , P is small and the explosive is almost certain to fail the test .sx It is useful to consider the probability of ignition which will almost certainly cause a device to be failed .sx To do this , it is necessary to fix a corresponding value for P ; that is , to give a numerical expression to the phrase " almost always " .sx If we define " reliable rejection " by requiring P 5% , we will obtain it whenever p p;r ; such that Similarly , for sufficiently small p , P approaches 1 and the explosive is almost certain to pass .sx So if we define " reliable acceptance " by requiring P 95% , we will obtain it whenever p p;a ; such that The range of possible p-values can thus be divided into three parts :sx Reliable rejection , P 5% , p;r ; p 1 Results not consistent , 5% P 95% , p;a ; p p;r ; Reliable acceptance , 95% P , 0 p p;a ; If we put these ranges side by side for different values of n , we obtain Fig. 1 , in which two curves of p;r ; against n ( Curve 1 ) and of p;a ; against n ( Curve 2 ) divide the area into three regions :sx consistent failures , results not consistent and consistent passes .sx