summary . 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 . it is suggested that tests requiring low ignition rates , and particularly no-ignition tests , are , as a class , poor discriminators . the ability to discriminate can be increased by increasing the number of ignitions accepted as the pass level . 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 . 1 . introduction . about a hundred million shots a year are fired in British mines and usually about 6 ignitions are reported each year . it is clear that with a practical ignition rate of roughly 10-7 , 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 . in rigorous terms this thesis demands that the ignition rate be multiplied ten million times or so . 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 &amp;sol; air , the absence of stemming in the shothole , and so on . ( 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 . 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 . however it is probably wise to assume that the contribution of the second group is substantial rather than preponderant . 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 . British approval tests have been such that an explosive is failed if ignitions are obtained in any of the tests . 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 ) . 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 . 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 can not be removed by any practicable increase in the number of shots fired . 2 . reliability and discrimination . a good test should meet , inter alia , two requirements : ( 1 ) . it should be reliable , i.e a repeat test of the same material should give the same result . ( 2 ) . it should have adequate discrimination , i.e it should indicate the presence of significant differences . no measurement is exactly reproducible , since all are subject to random errors . in explosive testing random error appears as a variation in the number of ignitions obtained in repeated tests on identical material . 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 . 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 . 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 . 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 . 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 . it is also fundamental that the acceptance and rejection limits can not be equal although , again , by firing enough shots they may be made to approach each other as closely as is desired . the difference between the acceptance and rejection levels is analogous to discrimination . 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 . this again is fundamental to all systems of assessment . it will be seen therefore that the concepts of reliability and discrimination as applied to testing are complex ones : overall , a system can be made reliable to a chosen extent at the limits of a chosen range . 3 . examination of the no-ignition test . 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 . 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 . correspondingly there should be a 0.95 probability of accepting an explosive at the acceptance level . calculations were then made which permitted the plotting of curve 1 in fig 1 . 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 . 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 . ) 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 . 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 . 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 . 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 . it appears to be impossible to avoid poor discrimination with no-ignition tests . 4 . tests permitting ignitions . in the last section it was found that poor discrimination appeared to be a characteristic of no-ignition tests : the effect of permitting one ignition is shown in fig 2 and fig 3 shows the characteristics for 2-ignition tests . 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 . the calculations on which fig 2 and 3 are based have been extended , and the results are summarized in table 1 . the accuracy of discrimination steadily increases with the number of ignitions ( m ) accepted as the pass level . confining attention for the time being to a reliable rejection level of pr equal to 0.1 , table 1 shows that the ratio ( pr/pa ) 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 . the table does not extend beyond the point where ( pr/pa ) 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 ( pr/pa ) for a given explosive fluctuates in the range of 2 to 1 . the following example may illustrate the operation of a test with a pass level of not more than 12 ignitions in 200 shots . this test has a reliable pr of 0.1 and a reliable pa ( 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 . what are the consequences of changing the sensitivity ? 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 . table 1 indicates that an economical and discriminating test at a rejection level of pr = 0.5 is to fire 35 shots and permit 12 ignitions . the calculations have since been extended by Mr G Fogg of S.M.R.E and it appears that at a rejection level of pr = 0.673 a discrimination ratio of 2 is obtained with a round ( n ) of 26 shots and a permitted number ( m ) of 13 ignitions . 5 . mathematical basis . 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 ) . 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 . for example , if our standard requirement is 0 ignitions in n trials , we have &amp;formula; . for sufficiently large p , P is small and the explosive is almost certain to fail the test . it is useful to consider the probability of ignition which will almost certainly cause a device to be failed . 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 failed . if we define reliable rejection by requiring P &lt; 5 % , we will obtain it whenever p &gt; pr such that &amp;formula; . similarly , for sufficiently small p , P approaches 1 and the explosive is almost certain to pass . so if we define reliable acceptance by requiring P &gt; 95 % , we will obtain it whenever p &lt; pa such that &amp;formula; . the range of possible p-values can thus be divided into three parts : reliable rejection , P &lt; 5 % , pr &lt; p &lt; 1 results not consistent , 5 % &lt; P &lt; 95 % , pa &lt; p &lt; pr reliable acceptance , 95 % &lt; P , 0 &lt; p &lt; pa . if we put these ranges side by side for different values of n , we obtain fig 1 , in which two curves of pr against n ( curve 1 ) and of pa against n ( curve 2 ) divide the area into three regions : consistent failures , results not consistent and consistent passes . 