βœ… 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 βˆͺ 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[XΒ²] = 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 = XΒ² with X uniform on {βˆ’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, Οƒ_temp = 5.66 and Οƒ_rides = 2.28. The Pearson correlation is r = Cov(X,Y) / (Οƒ_X Οƒ_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.

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