HW3: Pixel Quest Player Behavior

A Probability Distribution Investigation — Spring 2026 (Ch 3-4)

Author

Dr. Marcela Alfaro Córdoba

🎮 The Case

Welcome back, Statistical Detective!

Your probability work with StreamFlix made waves in the analytics community. Now you’ve caught the attention of GameZone, a mid-sized video game publisher launching indie titles on Steam, PlayStation, and Xbox.

The Chief Analytics Officer, Alex Rivera, needs your expertise:

“Our newest game, ‘Pixel Quest,’ has been live for 3 months. We have tons of player data—session lengths, purchase events, achievement unlocks—but we need someone who understands discrete probability distributions to help us model player behavior. We need to predict how many items players will buy, model achievement unlock patterns, and understand the randomness in player actions. Can you help us make sense of these patterns?”

Alex has data on 50,000 active players and needs you to apply discrete probability distribution models.

Your mission: Use discrete probability distributions to help GameZone understand and predict player behavior in Pixel Quest! Note that we will cover these topics during week 4.

Question 1: Discrete Random Variables Review

a. Define these terms in the GameZone context:

  • Random Variable:

  • Discrete Random Variable:

  • Probability Distribution:

  • Expected Value:

b. GameZone tracks the number of items a player purchases per session. Here’s the distribution:

Items Purchased (X) 0 1 2 3 4 5
Probability P(X) 0.30 0.25 0.20 0.15 0.07 0.03

Verify this is a valid probability distribution. Check both conditions:

Calculate:

  • \(E(X) = \mu =\) (show work)
  • \(Var(X) = \sigma^2 =\) (show work)
  • \(SD(X) = \sigma =\)
  • Interpret \(E(X)\) in the context of GameZone:

Question 2: Binomial Distribution — Achievement Unlocks

Historical data shows that 30% of players unlock the “Dragon Slayer” achievement on any given attempt.

a. Verify the BINS conditions for this scenario:

  • Binary outcomes:
  • Independence:
  • Number of trials fixed:
  • Same probability:

b. If 10 players attempt the achievement:

Use \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)

  1. \(P(X = 0)\) = (show work)
  2. \(P(X = 3)\) = (show work)
  3. \(P(X \leq 2)\) =
  4. \(P(X \geq 8)\) =

c. Calculate the mean and standard deviation:

  • \(\mu = np =\)
  • \(\sigma = \sqrt{np(1-p)} =\)
  • Interpret \(\mu\) in context:

d. GameZone offers a special reward if MORE THAN HALF of 10 players unlock the achievement in a day. What’s the probability of this happening?

Question 3: Geometric Distribution — First Purchase

Only 20% of new players make their first in-game purchase within their first session.

a. Explain why the Geometric distribution is appropriate here.

b. Let X = the number of sessions until a new player makes their first purchase.

  1. \(P(X = 1)\) = (first session purchase)
  2. \(P(X = 3)\) = (first purchase on 3rd session)
  3. \(P(X > 5)\) = (hasn’t purchased after 5 sessions)

Use: \(P(X = k) = (1-p)^{k-1} \cdot p\)

c. Expected number of sessions until first purchase:

\(E(X) = \frac{1}{p} =\)

Interpret this value for Alex:

d. Business decision:

If GameZone wants to target players who haven’t purchased yet with a special offer, after how many sessions should they send the offer? Justify using probability.

Question 4: Poisson Distribution — Bug Reports

On average, GameZone receives 4 bug reports per hour during peak gaming hours.

a. Explain why the Poisson distribution is appropriate:

b. Let X = number of bug reports in one hour. Using \(P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\):

  1. \(P(X = 0)\) =
  2. \(P(X = 4)\) =
  3. \(P(X \leq 2)\) =
  4. \(P(X > 6)\) =

c. Extended time periods:

If the rate is 4 per hour, what’s \(\lambda\) for a 30-minute window?

Calculate \(P(X = 2)\) for that 30-minute window.

d. Quality threshold:

GameZone considers more than 8 bug reports in one hour to be a “critical situation.” What’s the probability of a critical situation during peak hours?

Question 5: Comparing Distributions

GameZone is analyzing two different game features:

Feature A: 15 players are selected. Each has a 40% chance of using the new crafting system. (Use Binomial)

Feature B: On average, 6 players per hour discover the hidden Easter egg. (Use Poisson)

a. For Feature A:

  1. What distribution should you use? State the parameters.
  2. \(P(\text{exactly 6 players use crafting})\) =
  3. \(P(\text{more than 10 players use crafting})\) =
  4. \(E(X)\) and \(SD(X)\) =

b. For Feature B:

  1. What distribution should you use? State the parameter.
  2. \(P(\text{exactly 6 discover Easter egg in one hour})\) =
  3. \(P(\text{at least 1 discovers Easter egg in 30 minutes})\) =
  4. \(E(X)\) =

c. Comparison table:

Feature Distribution Mean Std Dev Key Assumption
A (Crafting)
B (Easter Egg)

Question 6: Revenue Modeling

GameZone sells “loot boxes” where each box independently has a 5% chance of containing a rare item.

a. Binomial setup:

A player buys 20 loot boxes. Let X = number of rare items obtained.

  1. State the distribution: \(X \sim B(n = \_\_, p = \_\_)\)
  2. \(P(X = 0)\) = (player gets no rare items)
  3. \(P(X \geq 2)\) =
  4. \(\mu =\) and \(\sigma =\)

b. Geometric setup:

Let Y = the number of loot boxes until the first rare item.

  1. \(P(Y = 1)\) =
  2. \(P(Y = 10)\) =
  3. \(E(Y)\) = On average, how many boxes until first rare item?

c. Business insight:

If each loot box costs $2.99, what’s the expected cost for a player to get their first rare item? Show your calculation.

d. Ethics discussion:

Some argue loot boxes are a form of gambling. Using the probability concepts from this homework, explain two arguments FOR and two arguments AGAINST this claim.

Question 7: Combined Analysis

A new Pixel Quest survey found: 65% of players prefer PvP (player vs. player) mode.

a. Sampling scenario:

GameZone randomly selects 12 players for a focus group.

  1. Let X = number of PvP-preferring players in the group. What’s the distribution?
  2. \(P(X = 8)\) =
  3. \(P(X < 4)\) =
  4. \(P(4 \leq X \leq 9)\) =

b. Quality assurance:

GameZone runs 8 randomly selected matches for testing. Each match independently has a 10% chance of having a connectivity error.

  1. \(P(\text{no errors in 8 matches})\) =
  2. \(P(\text{at least 2 errors})\) =
  3. \(E(\text{number of errors})\) =

c. Overall interpretation:

Write a brief (3-4 sentence) report to Alex Rivera summarizing what you found about player behavior using the distributions from this homework.

💭 Question 8: Detective’s Reflection

Reflect on your probability distribution investigation (5-7 sentences):

  • When would you use a Binomial vs. Geometric vs. Poisson distribution? What are the key differences?
  • How does the expected value help businesses make better decisions?
  • What surprised you about applying these distributions to gaming data?
  • How do discrete distributions differ from what you’ve seen with descriptive statistics?
  • Name one real-world business scenario (outside gaming) where each distribution would be useful.

🎉 Excellent work, Statistical Detective! GameZone now has the tools to model player behavior mathematically. Your discrete distribution skills have leveled up their analytics game!