# ✅ Checkpoint: Probability Foundations By Better Wise ## Quiz 1. In the sessions.csv table, P(mobile) = 0.40, P(converted) = 0.06, and P(mobile ∩ converted) = 0.018. What is P(mobile ∪ converted)? - 0.46 - 0.442 - 0.018 - 0.478 2. On the sessions table, 180 of 4,000 mobile sessions converted and 420 of 6,000 desktop sessions converted. Which statement correctly describes the relationship between device and conversion? - Dependent, because P(mobile) P(converted) = 0.024 differs from the joint probability 0.018. - Independent, because P(mobile | converted) = 0.30 is computed from the same joint count. - Independent, because P(converted | mobile) = 0.045 and P(converted | desktop) = 0.07 are both small. - Dependent, because the two events are disjoint. 3. Which of the following statements about the 8-row order_value extract (values 20, 40, 10, 0, 30, 0, 50, 10) are correct? Select all that apply. - The population variance is 300 dollars. - The standard deviation is about 17.32 dollars. - E[X] = 20 dollars. - The mean of 20 is a value that every session in the file actually produced. - E[X²] = 700 dollars squared. 4. A colleague reports Cov(X, Y) = 0 for two columns and concludes they are independent. Which points correctly explain why this conclusion can fail? Select all that apply. - If Y = X² with X uniform on {−1, 0, 1}, then Cov(X, Y) = 0 while Y is fully determined by X. - Zero covariance is impossible unless the two columns are independent. - Covariance of zero means the two columns must have equal variances. - Independence implies zero covariance, but zero covariance does not imply independence. - Covariance only detects linear association, so a symmetric curved relationship can give zero covariance. 5. On the bike-share data, Cov(temp, rides) = 12.8, σ_temp = 5.66 and σ_rides = 2.28. The Pearson correlation is r = Cov(X,Y) / (σ_X σ_Y), which here is approximately ____. 6. For any constants a and b, the variance of aX + b equals ____. 7. A correlation near zero means there is no ____ association between the two columns; it does not prove independence. 8. A dashboard reports a mean order value of 20 dollars with a standard deviation of 17 dollars. How should a reader interpret this? - Every session brings in close to 20 dollars, so 0-dollar sessions indicate a data error. - The mean of 20 must be one of the observed order values. - The variance is 17 dollars squared, so the spread is small relative to the mean. - Orders of 0 and 50 dollars are ordinary variation, not outliers. 9. Two events A and B are disjoint and both have positive probability. What is their dependence status? - Independent only if P(A) = P(B). - Dependent, because P(A ∩ B) = 0 while P(A) P(B) > 0. - Neither dependent nor independent, since the product rule does not apply. - Independent, because they cannot occur together. 10. Take a dataset question of your own, for example 'do mobile visitors convert less?'. Write out the sample space, name the events, state whether you need a marginal or a conditional probability, and test independence with the multiplication rule. Then explain in two or three sentences why the conditional probability answers a sharper question than the marginal one. --- ## Contents Part of [🎲 Mathematics for Machine Learning 3: Probability and Statistics for Data-Driven Models](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data.md) - **🎲 Probability Foundations for Data** - [🧩 Axioms, Conditional Probability and Independence](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/probability-axioms-conditional-independence.md) - [📊 Expectation and Variance on a Real Dataset](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/expectation-variance-worked-dataset.md) (sign-in required) - [🔗 Covariance, Correlation and Dependence](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/covariance-correlation-dependence.md) - [✅ Checkpoint: Probability Foundations](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/checkpoint-probability-foundations.md) — this lesson - **📈 Distributions and Bayes' Rule** - [🔢 Bernoulli, Binomial and Poisson Counts](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/bernoulli-binomial-poisson-count-data.md) - [🔔 Gaussian and Exponential Models](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/gaussian-exponential-continuous-data.md) - [🧠 Bayes' Rule and Base Rates](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/bayes-rule-base-rates.md) - [🧪 Checkpoint: Distributions and Bayes](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/checkpoint-distributions-and-bayes.md) - **🎯 Estimation and Inference** - [⚙️ Maximum Likelihood and MAP Estimation](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/maximum-likelihood-map-estimation.md) - [⚖️ Bias-Variance Tradeoff in Practice](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/bias-variance-tradeoff-practice.md) - [📐 Sampling, Confidence Intervals and Hypothesis Tests](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/sampling-confidence-intervals-hypothesis-tests.md) (premium content) - [🏁 Checkpoint: Estimation and Inference](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/checkpoint-estimation-and-inference.md) (premium content) - [Course cheatsheet](https://everwise.study/courses/mathematics-for-machine-learning-3-probability-and-statistics-for-data/cheatsheet.md) (premium content)