Mathematics Behind Case Opening Simulator

📊 Expected Value (EV) Calculation

The expected value of opening a case is calculated using probability theory and represents the average amount you would receive if you opened the case an infinite number of times.

Basic Formula

🧮 Core EV Formula

EV = Σ(probability_i × value_i)

probability_i = chance of getting item i

value_i = market value of item i

The probability for each item is calculated as:

probability_i = rarity_chance / items_in_rarity

Special Cases: Blue Gems

For Case Hardened items that can be Blue Gems, we adjust the expected value calculation to account for pattern probabilities:

💎 Blue Gem EV Formula

EV_{case_hardened} = (non_blue_patterns/1000 × base_value) + Σ(tier_patterns_i/1000 × tier_value_i)

non_blue_patterns = number of non-blue gem patterns (out of 1000)

tier_patterns_i = number of patterns in tier i

tier_value_i = value of blue gems in tier i

📈 Variance and Risk Calculation

To understand the risk and calculate required bankrolls, we need to know how much the returns can vary from the expected value.

📊 Risk Calculation Formulas

Variance = Σ(probability_i × (value_i - EV)²)

The standard deviation (σ) is then:

σ = √Variance

💰 Required Bankroll Calculation

The required bankroll is calculated to ensure you have enough money to avoid bankruptcy with a specified confidence level.

📈 Mathematical Model

Total Profit ~ N(n×μ, n×σ²)

μ = net expected value per case

σ = standard deviation per case

n = number of cases

🚫 Bankruptcy Avoidance

P(Total Profit > -Bankroll) = C

This leads to:

n = (z×σ/μ)²

z = z-score for confidence level C

Required bankroll = n × case_price

🎯 Bankroll Confidence Calculation

Given a user's bankroll, we calculate the probability they won't go bankrupt.

🔢 Confidence Calculation Process

For a given bankroll B that allows n cases:

n = B/case_price

The z-score for bankruptcy avoidance is:

z = (n×μ)/(√n×σ)

Confidence = Φ(z) × 100%

Where Φ is the standard normal cumulative distribution function.

🎨 Float Values and Price Adjustment

Each item has a float value (wear) that affects its price. We calculate average prices by considering the distribution of possible float values and their corresponding price impacts.

Float Ranges by Category

🧤 Gloves

0.06 - 0.80

🔪 Special Knives

(Karambit, Doppler, Fade)

0.00 - 0.08

🩸 Rust Coat Knives

0.40 - 1.00

⚔️ Other Knives

0.00 - 0.80

🔫 Regular Items

0.00 - 1.00

Wear Categories

Factory New (FN)

< 0.07

Minimal Wear (MW)

0.07 - 0.15

Field-Tested (FT)

0.15 - 0.38

Well-Worn (WW)

0.38 - 0.45

Battle-Scarred (BS)

0.45 - maxFloat

Price Calculation

🧤💎 Gloves (High Value > $1000)

Price multipliers by wear:

FN: 1.0× base value

MW: 0.25× base value

FT: 0.154× base value

WW: 0.143× base value

BS: 0.125× base value

🧤 Gloves (Regular Value ≤ $1000)

FN: 1.0× base value

MW: 0.4× base value

FT: 0.333× base value

WW: 0.286× base value

BS: 0.25× base value

🔫 Regular Items

For float value f, price is calculated as:

FN: base × (1 - f/4)

MW: base × (1 - f/5) / 1.1

FT: base × (1 - f/6) / 1.2

WW: base × (1 - f/7) / 1.3

BS: base × (1 - f/8) / 1.5

Average Price Calculation

📊 Average Price Calculation Process

1

Determine the item's float range based on its category

2

Split the float range into wear categories

3

Calculate the share (probability) of each wear category:

share = (category_max - category_min) / (max_float - min_float)

4

For each wear category:

• Calculate the midpoint float value

• Calculate the price at that float value

• Multiply by the category's share

5

Sum all weighted prices to get the final average price

avg_price = Σ(share_i × price(midpoint_i))

share_i = probability of wear category i

midpoint_i = middle float value in category i

price() = price function based on item type

⚠️ Assumptions and Limitations

🎲 Independence

Each case opening is assumed to be independent of previous openings.

📊 Normal Distribution

For large numbers of cases, we assume the total profit follows a normal distribution (Central Limit Theorem).

💰 Market Stability

Calculations assume item values remain relatively stable during the case opening session.

🔢 Pattern Distribution

For Case Hardened items, we assume pattern numbers are uniformly distributed.