Calculate conditional probability for coin flips

Question

Given the following scenario about revenue per session, calculate the the sample size needed.

Codemia · Accepted Answer

We need to calculate the conditional probability related to the revenue per session from coin flips in a statistical experiment. Let's denote the random variable $ X $ as the revenue per session, which can take values based on the outcome of coin flips. The outcome of a fair coin flip can be represented as $ Y $, where $ Y = 1 $ indicates heads (success) and $ Y = 0 $ indicates tails (failure). We are interested in finding the conditional probability $ P(X | Y) $, which describes the distribution of revenue given the outcome of the coin flip.

To calculate the required sample size for a given confidence level and power, we use the formula derived from the central limit theorem. Let's assume that the revenue per session follows a normal distribution. We can use the following steps:
1. Define the significance level $ \alpha $ (commonly 0.05) and the power $ (1 - eta) $ (commonly 0.8).
2. Estimate the effect size $ d $ based on prior data or pilot studies, which is the expected difference in means between the two groups (e.g., revenue from heads vs. tails).
3. Use the formula for sample size calculation:
   
   $$ n = \frac{(Z_{\alpha/2} + Z_{\beta})^2 \cdot (\sigma_1^2 + \sigma_2^2)}{d^2} $$
   
   Here, $ Z_{\alpha/2} $ and $ Z_{\beta} $ are the Z-values corresponding to the chosen significance level and power, and $ \sigma_1 $ and $ \sigma_2 $ are the standard deviations of the two groups.

Suppose from historical data, we estimate the following: 
- The revenue per session when the outcome is heads has a mean $ \mu_1 = 100 $ and standard deviation $ \sigma_1 = 20 $.
- The revenue per session when the outcome is tails has a mean $ \mu_2 = 80 $ and standard deviation $ \sigma_2 = 25 $.
- We want to detect an effect size $ d = \mu_1 - \mu_2 = 20 $.
- For $ \alpha = 0.05 $ (two-tailed), $ Z_{\alpha/2} \approx 1.96 $ and for power $ (1 - \beta) = 0.8 $, $ Z_{\beta} \approx 0.84 $.

Plugging in the values:
$$ n = \frac{(1.96 + 0.84)^2 \cdot (20^2 + 25^2)}{20^2} = \frac{(2.8)^2 \cdot (400 + 625)}{400} = \frac{7.84 \cdot 1025}{400} = 20.1 $$

Rounding up, we find we need a sample size of at least 21 sessions for each group.

The result indicates that to achieve a statistically significant result with a power of 80% and a significance level of 5%, we need to conduct at least 21 coin flips leading to heads and 21 leading to tails. This sample size ensures that we can reliably detect differences in revenue per session attributed to the outcome of the coin flips, taking into account the variability in revenue.

This analysis can be extended to discuss multiple testing corrections if we were to test several hypotheses simultaneously. For example, if we were to compare more than two variations of revenue strategies based on different coin flip outcomes, we could employ the Bonferroni correction or FDR to adjust the significance level. Additionally, we might consider Bayesian methods to incorporate prior information about revenue distributions, which could refine our estimates for sample size and improve decision-making in a business context.

Calculate conditional probability for coin flips

DE Shaw

What the Interviewer Expects

Key Topics to Cover

How to Approach This

Possible Follow-up Questions

Sharpen Your Skills on Codemia

Sample Answer

Problem Formulation

Solution Approach

Submit Your Answer

DE Shaw Data Scientist Interview Guide

Related Questions

Interpret mixed results across segments from an experiment

Explain instrumental variables with an example

Calculate conditional probability for coin flips

Explain difference-in-differences with an example

Explain p-value with an example