Consistent Purchase Probability

Solution & Explanation

Understanding the Problem:

The problem involves estimating the long-term probability of an item "A" being purchased, given that it has been recommended 10 times and purchased 10 times. The context assumes that the likelihood of a user buying any product is evenly distributed across all items.

Key Points:

1. Uniform Probability Assumption:

   • The problem states that the probability of users buying each item is uniform, suggesting each item has the same baseline probability of being purchased when recommended.

2. Current Observations for Item A:

   • Item "A" has been recommended 10 times and purchased 10 times, leading to an empirical probability of 1 (100%) based on this limited data.

3. Long-Term Probability Estimation:

   • While the observed probability is 100%, it's crucial to estimate the long-term probability by considering the potential variability and uncertainty due to the small sample size.

Bayesian Estimation Approach:

To overcome the overconfidence in the empirical probability due to a small sample size, we can apply Bayesian inference to estimate the long-term probability.

1. Beta Distribution as a Prior:

   • Use a Beta distribution, which is parameterized by two parameters, $\alpha$ (successes) and $\beta$ (failures).
   • Start with a non-informative prior, Beta(1, 1), which assumes all probabilities are equally likely.

2. Updating with Observed Data:

   • After observing 10 successes (purchases) and 0 failures (non-purchases), update the Beta distribution: $\alpha = 1 + 10 = 11$ $\beta = 1 + 0 = 1$
   • The posterior distribution becomes Beta(11, 1).

3. Calculating the Long-Term Probability:

   • The mean of the Beta distribution (interpreted as the long-term probability) is given by: $\text{Mean of Beta}(\alpha, \beta) = \frac{\alpha}{\alpha + \beta}$
   • For Beta(11, 1), the mean is: $\frac{11}{11 + 1} = \frac{11}{12} \approx 0.917$
   • This suggests that the long-term probability of item "A" being purchased is approximately 91.7%.

Considerations:

• Sample Size Effect:
  • With only 10 recommendations, the sample size is small, and the estimate has high uncertainty. As more data is collected, the estimate will stabilize.
• Real-World Variability:
  • In practice, the probability of purchase may fluctuate due to factors like changes in user preferences, item popularity, and market conditions.

Conclusion:

By using Bayesian estimation, we account for the small sample size and avoid overconfidence in the empirical probability. The estimated long-term probability of item "A" being purchased, given the data, is approximately 91.7%, reflecting a more realistic expectation of future outcomes.