## 4 Results
### Analysis of Monte Carlo Simulations  
Figure 11 presents an exemplary MCS output for three of the six drivers. The plots show the fraction of races that the drivers have completed on the respective positions. They tell us about the expected race outcome with a given race strategy, which can be used to evaluate different variants. The first row in the figure (w/o MCS) displays the deterministic simulation output, that is, without using MCS. The second and third rows (w/MCS) show the resulting position distributions for two different race strategies for [PERSON] when using MCS. In this example, [PERSON]'s pit stop was postponed from lap 14 (strategy 1) to lap 19 (strategy 2). We want to point out three aspects of the figure. Firstly, the second row provides much more information than the first row because MCS was applied. For [PERSON], for example, it turns out that fifth place in the race is only the second most likely outcome, although the deterministic simulation shows this as a result. For [PERSON], it is almost as likely to finish second as it is to finish first. Secondly, we can observe a shift in the position distribution of [PERSON] when switching from strategy 1 to strategy 2. It results in an improvement in third positions. This would again not have been visible in the deterministic simulation. Similar investigations can also be carried out with different tire compounds, for example. Thirdly, the fraction of rank position six is slightly above the previous ones for [PERSON]. Retirements due to accidents and failures explain this. Hence, MCS allows us to determine a basic race strategy before a race that already considers probabilistic influences. At the same time, we gain an idea of the robustness of the strategy against unforeseen events because it is fixed, that is, not adapted to the race situation during the simulation. During a real race, of course, this basic strategy must be adapted to the current situation, for example, SC phases. Without MCS, we would only obtain a single result, which is much less helpful for strategy determination. Figure 11: Resulting position distributions for three of the six simulated drivers in an exemplary race without using Monte Carlo simulation (MCS) (w/o MCS) and with using MCS (w/MCS, 10,000 simulation runs). The pit stop of [PERSON] was postponed from lap 14 (strategy 1) to lap 19 (strategy 2) in this example. Driver abbreviations: [PERSON]–[PERSON], VET–[PERSON], [PERSON]–[PERSON]. It is not only final positions that can be evaluated with the combination of race simulation and MCS. Figure 12 depicts the distribution of race durations after 10,000 simulation runs for this exemplary race. The durations of races without an SC phase differ only slightly. The widely distributed hill between 95 min to 103 min indicates races with a single SC phase. Races with two and three SC phases also have a high spread but appear relatively seldom. However, the fraction of SC phases in the simulation fits well with the actual data. This is outlined in Table 7. The VSC fractions differ slightly due to the conditional probability depending on previous failures, cp. Equation (15). But how reliable is the MCS result after, for example, 10,000 simulation runs? According to the law of large numbers, it approaches the expectation ever more closely, the more often the random experiment is repeated. Table 8 shows an evaluation of [PERSON]'s mean rank position in an exemplary race after 20 batches of simulation runs. It can be seen that the deviation between the batches decreases with a rising number of simulation runs per batch. This behavior is similar for all drivers. In most cases, 10,000 simulation runs offer a good compromise between computation time and certainty. However, it must be emphasized that the result of the race itself depends strongly on a correct parameterization of the race simulation.