A pharmaceutical company is testing a new drug's effectiveness by measuring various biochemical markers and patient responses, classifying outcomes as either "Effective" or "Not Effective." The probability of the drug being effective is chosen independently with equal probability for each test.
The company aims to build a classification model to predict drug effectiveness. They are especially concerned with the cost of false negatives (predicting the drug is ineffective when it is effective) and require a model that is interpretable, reliable, and generalizes well across different data subsets.
Given these requirements, which metrics would be most appropriate for evaluating the performance of this classification model?
- p-value to determine the statistical significance of the model's predictions in the tests.
- Type I error to determine the rate of false positives in the model's classifications.
- R-squared (R²) indicates the proportion of variance the model explains across the dataset.
- t-statistic to assess the difference between the predicted values and actual test outcomes.
- Cross-validation scores to evaluate model performance and generalizability across different data subsets.
In the enchanted forest of Eldoria, there are 225 mystical trees, all initially dormant. The forest guardian performs a special ritual every night for 225 nights. On each night n (where 1 ≤ n ≤ 225), the guardian toggles the state of every n-th tree (i.e., trees at positions n, 2n, 3n,...).
Additionally, on every 30th night, the guardian casts a powerful enchantment that locks the state of every tree that is currently active, preventing it from being toggled again for the rest of the 225 nights.
How many trees will be active at the end of the 225 nights?
Let Y be a random variable over the nonnegative integers such that the probability of Y taking a value n is given by:
P(Y = n) = (3/2) × (1/2)^n
What is the probability that Y takes on an odd value not greater than 5?
Options:
- 1/486
- 9/54
- 1/54
- 1/6
- 91/486
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