Practice — Probability Fundamentals (6 questions)

A payments team deploys a fraud model. Historically, 0.2% of transactions are fraudulent. The model flags 95% of fraudulent transactions (recall / sensitivity) and incorrectly flags 2% of legitimate transactions (false-positive rate).

1. A transaction is flagged. What is the probability it is actually fraudulent? Show the calculation.
2. The team is disappointed and asks whether improving recall from 95% to 99% would fix the problem. Would it? What lever actually matters here?
3. The same model is deployed on a merchant segment where 5% of transactions are fraudulent. What is the precision of a flag now?

For each scenario, name the most appropriate distribution, state its parameter(s) from the information given, and compute the quantity asked.

1. A checkout page has a 4% conversion rate. 50 independent visitors arrive. What is the expected number of conversions and the standard deviation of that count? Roughly what is the probability of exactly zero conversions?
2. A support inbox receives on average 2 urgent tickets per hour, arriving independently. What is the probability of receiving no urgent tickets in a given hour? At least 3?
3. Servers in a fleet fail at a constant rate with a mean time between failures of 500 hours. A server has already run for 200 hours without failing. What is the probability it survives another 100 hours?
4. Which of the above would you not model with a Normal distribution, and why?

Your product has a baseline sign-up rate of 8%. A colleague runs an experiment with 2,000 users in control and 2,000 in treatment and reports "treatment converted at 8.9% vs 8.0% — an 11% relative lift, ship it!"

1. Using the CLT, compute the standard error of each arm's conversion rate and the standard error of the difference between arms.
2. How many standard errors is the observed 0.9 percentage-point difference? What does that say about whether the lift is real?
3. The colleague proposes fixing this by running the test on 4× as many users. By what factor does the standard error shrink? What if they wanted the standard error to be 10× smaller?
4. Explain in one or two sentences why the CLT — not the distribution of individual sign-ups — is what justifies these calculations.

An analytics team studies whether users who enable dark mode are more likely to be paying subscribers. Out of all users, 40% use dark mode. The joint proportions are:

1. Compute the marginal probability of paying and the conditional probabilities P(\text{Paying} \mid \text{Dark}) and P(\text{Paying} \mid \text{Light}). Are dark-mode usage and paying status independent?
2. A second analyst computes the Pearson correlation between a user's daily session count X and a feature Y = (X - \bar{X})^2 (squared deviation from the mean) and finds it is approximately zero. They conclude the two are unrelated. What is wrong with this conclusion?
3. Two microservices each have a 1% chance of failing on a given day. An engineer estimates the chance both fail on the same day as 0.01 × 0.01 = 0.0001. What assumption is being made, and give a concrete reason it could be badly wrong.

1. Show why 23 people are enough for a better-than-even chance that two share a birthday. Set up the complement, and explain intuitively why the answer is so much smaller than 365/2.
2. A team generates 6-character IDs from a 36-character alphabet (36^6 ≈ 2.18 billion possibilities) and assumes collisions are "practically impossible" until they have billions of records. Using the birthday approximation P(\text{collision}) \approx 1 - e^{-n^2 / (2H)}, estimate the number of IDs at which the collision probability reaches 50%. Is the assumption right?
3. Explain the Monty Hall result (switching wins with probability 2/3), and state exactly which assumption about the host the result depends on.

A ride-hailing app charges a base fare plus a per-minute rate. Trip duration T (minutes) has mean 18 and standard deviation 6. The fare is F = 2.50 + 0.40\,T dollars.

1. Compute E[F] and \mathrm{SD}(F).
2. A driver completes 40 trips in a day. Assuming trips are independent, what are the mean and standard deviation of the driver's daily fare total? Which of these two answers still holds if trips are not independent, and why?
3. A promotion gives each of 1,000 riders an independent 3% chance of a free ride. Using indicator variables, compute the expected number of free rides and its variance. Then explain why the expected count would be unchanged even if the 3% chances were strongly correlated across riders (e.g. all triggered by the same weather event), but the variance would not.