One of the most difficult questions for self-driving cars concerns their safety: How can we determine whether a particular self-driving car model is safe? The most popular an­swer to this question is based on a straightforward application of statis­tics and leads to conclusions such as that “…fully autonomous vehicles would have to be driven hundreds of millions of miles and sometimes hun­dreds of billions of miles to demon­strate their reliability…”. This state­ment comes from a recent RAND re­port by Nidri Kalra and Susan Pad­dock on the topic. Unfortunately, these statements are untenable in this form because the statistical argument contains major oversights and mis­takes, which we will point out in the following.

7.1 Failure rate estimation

The argument is usually presented as a problem of failure rate estimation where observed failures (accidents involving self-driving cars) are com­pared against a known failure rate (accident rates of human drivers). Accidents are modeled as discrete, independent and random events that are determined by a (statistically con­stant) failure rate. The failure rate for fatal accidents can be calculated by dividing the number of accidents with fatalities by the number of vehi­cle miles traveled. If we consider the 32,166 crashes with fatalities in traf­fic in the US in 2015 and relate them to the 3.113 billion miles which mo­tor vehicles traveled, then the failure rate is 32,166 / 3.113 billion = 1.03 fatalities per 100 million miles. The probability that a crash with fatality occurs on a stretch of 1 mile is ex­tremely low (0,0000010273%) and the opposite, the success rate, the probability that no accident with fa­tality occurs on a stretch of 1 vehicle-mile-traveled (VMT) is very high (99,999998972%). By observing cars driving themselves, we can obtain es­timates of their failure rate. The con­fidence that such estimates reflect the true failure rate increases with the number of vehicle miles traveled. Simple formulas for binomial proba­bility distributions can be used to cal­culate the number of miles which need to be driven without failure to reach a certain confidence level: 291 million miles need to be driven by a self-driving car without fatality to be able to claim with a 95% confidence level that self-driving cars are as reli­able as human drivers. This is nearly three times the distance between fa­talities that occur during human driv­ing. If we relax the required confi­dence level to 50%, then at least 67 million miles need to be driven with­out fatality before we can be confi­dent that self-driving cars are safe. Although this calculation is simple most authors – including the authors of the RAND report – use the wrong measures. Instead of dividing the number of crashes involving fatalities (32,166) by VMT, they divide the number of fatalities (35,091) by VMT. This overstates the failure rate of human drivers because a single ac­cident may lead to multiple fatalities and the number of fatalities per fatal accident may depend on many fac­tors other than the reliability of the driver.

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