Model wins-until-failure and expected future wins

You are given a DataFrame df where each row summarizes a player’s performance until their first loss.

Input

df columns:

• player_id (string/int)
• wins_before_first_loss (int, $k\ge 0$ ): number of consecutive wins observed before the first loss occurred

Interpretation: for each player, you observed a sequence of games that ended with a loss, e.g., WWW...WL, and wins_before_first_loss = k.

Questions

1. Compute the expected number of additional wins before the next loss for each player.
   • You may assume each player has an underlying win probability $p_i$ that is constant across games.
   • Show how you would compute this expectation from the observed data (code/pseudocode is fine).
2. Propose a probabilistic model for wins_before_first_loss and explain how you’d estimate parameters.
3. Given a plot of the computed expectations across players (distribution/shape), interpret what it suggests (heterogeneity, outliers, model misfit).
4. How would you evaluate the model and use it to make a decision (e.g., ranking players, allocating resources, setting thresholds)?