Approximating small probabilities using importance sampling

Box plots are often used. They are not always the best visualisation (e.g. is bi-modality not visible), but I will not discuss that in this post. Here, I will use it as an example of Importance sampling that is a technique to estimate tail probabilities without requiring many, many simulations. Assume that we have a standard normal distribution. What is the probability of getting a box plot extreme outlier (here we use that it must be greater than 2.