The first part I handled fine, it's just plugging into the conditional directly since gender is observed.
First, clarify that the overall conversion rate is a weighted average of the gender-specific rates, so the male rate (3%) is the updated probability after learning the user is male. Then, for the reverse, apply Bayes' theorem using the prior gender distribution implied by the overall rate, or state that it cannot be determined without the proportion of male users.
Pro tip: Always check if the prior gender mix is given; if not, explicitly state the assumption (e.g., 50/50) and show how the answer changes. This demonstrates rigor and avoids silent errors.
Identify the overall conversion rate (2%) and the conditional rates for male (3%) and female (1%). Recognize that the overall rate is a weighted average of these conditional rates.
Since the user is male, the updated conversion probability is simply the male conversion rate, 3%. No further calculation is needed because the condition is directly given.
Use the overall rate and conditional rates to solve for the proportion of male users. Let p be the proportion male: 0.02 = 0.03p + 0.01(1-p). Solve to find p = 0.5 (assuming the rates are accurate).
Compute P(Male | Conversion) = P(Conversion | Male) * P(Male) / P(Conversion) = (0.03 * 0.5) / 0.02 = 0.75. So there is a 75% chance the user was male given a conversion.
Note that the calculation relies on the overall rate being exactly the weighted average and on the gender proportions being 50/50. If the proportions are unknown, the reverse probability cannot be determined without additional information.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.