Some Code and Dataseta for Probability Theory & Mathematical Statistics.

The card draw system, a direct factor behind the huge success of Genshin Impact, is explored to provide players with suggestions for better asset planning and probability understanding. First, we introduce the three common types of card draws: character, weapon, and resident, as well as the official probabilities and rules of card draws. Then we analyze the probability of card draw statistically, and discover some hidden mechanisms, such as the guaranteed mechanism, the probability UP mechanism, the divine casting trajectory mechanism, and the smoothing mechanism. Then a model based on the roulette wheel selection method was built, and the probability distribution of drawing different star ratings and categories of items was calculated using the dynamic programming algorithm, and compared with the statistical results to verify the validity of the model. Finally, some conclusions and inferences are drawn from the model, such as the expected number of draws for drawing UP characters and weapons, the optimization of card drawing strategy, the estimation of card drawing cost, etc. The shortcomings of the model and the future work are also explained.