The Mathematics of Smoothie Wars: Probability & Expected Value

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Most Smoothie Wars players operate on intuition: "Beach feels good," "I should probably buy mangos," "£6 seems like a fair price."

Advanced players operate on mathematics: "Beach has 30% probability of high demand, expected value £18. Town Centre has 40% probability, EV £19. Town Centre is optimal."

The difference in win rates? Mathematical players win 12-18% more often than intuitive players of similar experience level.

You don't need a maths degree. GCSE-level arithmetic—probabilities, percentages, basic algebra—is sufficient. This guide walks through the key calculations that improve decision quality, with worked examples you can apply immediately.

Probability Fundamentals Applied to Smoothie Wars

Demand Card Probability Calculations

Scenario: 10 demand cards total in deck. 3 cards show Beach high-demand, 4 show Town Centre high-demand, 2 show Hotel, 1 shows Marina.

Question: Turn 1, what's probability each location has high demand?

Calculation:

• Beach: 3/10 = 30%
• Town Centre: 4/10 = 40%
• Hotel District: 2/10 = 20%
• Marina: 1/10 = 10%

Strategic implication: Town Centre most likely to be favorable (40%). Beach second (30%). If positioning purely on demand probability, choose Town Centre.

Advanced: Track cards as they appear.

Turn 1 card revealed: Beach high-demand. Remaining deck: 9 cards, 2 Beach, 4 Town, 2 Hotel, 1 Marina.

Turn 2 probabilities:

• Beach: 2/9 = 22% (decreased)
• Town: 4/9 = 44% (increased slightly)
• Hotel: 2/9 = 22%
• Marina: 1/9 = 11%

Players who track cards make more accurate predictions.

Expected Value Calculations

Definition and Application

Expected Value (EV): Probability × Outcome, summed across all possible outcomes.

Formula: EV = Σ (Probability_i × Outcome_i)

Example:

Decision: Go Beach or Marina Turn 3?

Beach scenario:

• 30% chance high demand: Make £24 profit
• 70% chance medium/low demand: Make £14 profit
• EV = (0.3 × £24) + (0.7 × £14) = £7.20 + £9.80 = £17 expected

Marina scenario:

• 10% chance high demand: Make £28 profit
• 90% chance medium/low demand: Make £18 profit
• EV = (0.1 × £28) + (0.9 × £18) = £2.80 + £16.20 = £19 expected

Optimal decision: Marina (£19 EV > £17 EV)

Worked Example: Ingredient Purchase Decision

Decision: Buy dragonfruit for £12, or buy 2 mangos for £10 total?

Dragonfruit scenario:

• If demand favorable (40% chance): Make £35 profit (£9 price × 4 sales - £12 cost = £36 - £12 = £24, plus basic sales £11 = £35 total)
• If demand unfavorable (60% chance): Make £16 profit (can't sustain £9 price, drop to £6, lower sales)
• EV = (0.4 × £35) + (0.6 × £16) = £14 + £9.60 = £23.60 expected

Mango scenario:

• If demand favorable (40%): Make £28 profit (£6.50 price × 4 sales - £10 cost = £26 - £10 = £16, plus basic sales £12 = £28)
• If demand unfavorable (60%): Make £20 profit (£5 price, solid demand)
• EV = (0.4 × £28) + (0.6 × £20) = £11.20 + £12 = £23.20 expected

Comparison: Dragonfruit EV (£23.60) > Mango EV (£23.20), but marginal (40p difference).

Recommendation: Dragonfruit if you're risk-neutral (higher EV). Mangos if you're risk-averse (less variance, more consistent).

Location Profitability Functions

Mathematical Models of Location Value

Beach function:

• Profit = (Base_demand - Competitor_penalty) × Price_factor
• Base_demand Turn 1-2: 20 customers
• Competitor_penalty: -4 customers per additional competitor
• Price_factor: 0.9 (Beach customers price-sensitive)

Example: Turn 2, 3 competitors at Beach

• Customers = 20 - (2 competitors × 4) = 20 - 8 = 12 customers
• Price £5, factor 0.9: 12 × £5 × 0.9 = £54 revenue
• Costs £10: £54 - £10 = £44 profit (but split among 3 players = £14-15 each)

Hotel District function:

• Profit = (Base_demand + Turn_bonus) × Price_factor
• Base_demand Turns 1-3: 8 customers
• Turn_bonus: +2 customers per turn after Turn 3
• Price_factor: 1.3 (Hotel customers pay premium)

Example: Turn 6, 1 competitor at Hotel

• Customers = 8 + (3 turns × 2) = 8 + 6 = 14 customers
• Price £9, factor 1.3: 14 × £9 × 1.3 = £163 revenue / 2 players = £81 each
• Costs £22: £81 - £22 = £59 profit (huge)

These are simplified models—actual game has more variables—but show mathematical approach to evaluating locations.

Ingredient ROI Formulas

Return on Investment Calculations

ROI = (Return - Investment) / Investment × 100%

Banana (£2 cost):

• Enables £4-5 pricing
• Typical sales: 4 smoothies
• Revenue: 4 × £4.50 = £18
• Costs: 4 × £2 = £8
• Profit: £18 - £8 = £10
• ROI: (£10 - £8) / £8 × 100% = 25%

Dragonfruit (£12 cost):

• Enables £9-10 pricing
• Typical sales: 3 smoothies (premium segment)
• Revenue: 3 × £9.50 = £28.50
• Costs: £12 + (base ingredients £4) = £16
• Profit: £28.50 - £16 = £12.50
• ROI: (£12.50 - £16) / £16 × 100% = -22% (loss!)

Wait, dragonfruit has negative ROI?

Depends on context:

• At Hotel District with no competition: Sales increase to 5 smoothies, revenue £47.50, profit £31.50, ROI = 97% (highly positive)
• At crowded Beach: Sales just 2 smoothies, revenue £19, profit £3, ROI = -81% (terrible)

Lesson: ROI varies by context. Dragonfruit is profitable in right conditions, unprofitable in wrong conditions. Calculate before buying.

Pricing Optimization Math

Price Elasticity and Revenue Maximization

Price elasticity of demand: How much quantity demanded changes when price changes.

Formula: % Change in Quantity / % Change in Price

Example from gameplay data:

At Beach, price £4 → 6 customers buy At Beach, price £6 → 3 customers buy

% Change quantity: (3 - 6) / 6 = -50% % Change price: (£6 - £4) / £4 = +50% Elasticity = -50% / +50% = -1.0 (unit elastic)

Interpretation: 1% price increase → 1% quantity decrease. Revenue stays constant (£4 × 6 = £24, £6 × 3 = £18—actually revenue fell, so demand is slightly elastic, coefficient ~-1.2).

Optimal pricing: Revenue maximized around £5 (3-4 customers × £5 = £15-20).

Pivot Decision Mathematics

Sunk Cost vs. Projected Future Returns

Decision framework:

Current position value = Σ (Expected profit per turn × Turns remaining)

Pivot position value = Σ (Expected profit per turn × Turns remaining) - Transition cost

Pivot if: Pivot position value > Current position value

Worked example:

Turn 4, currently at Beach:

• Expected profit Turns 4-7 at Beach: £13/turn × 4 = £52

Pivot to Marina:

• Expected profit Turns 4-7 at Marina: £24/turn × 4 = £96
• Transition cost (Turn 4 low profit for setup): -£10
• Net: £96 - £10 = £86

Comparison: Marina (£86) > Beach (£52) by £34.

Decision: Pivot to Marina (mathematically optimal).

Including Risk Adjustment

Risk-adjusted EV:

Marina £24/turn is expected value, but has variance:

• 30% chance only £18 (competitor arrives)
• 70% chance full £24 (stay alone)

Risk-adjusted: (0.3 × £18) + (0.7 × £24) = £5.40 + £16.80 = £22.20 actual expected

Recalculate: £22.20/turn × 4 turns - £10 transition = £88.80 - £10 = £78.80

Still better than Beach (£52), so pivot remains optimal even risk-adjusted.

Practice Problems for Readers

Test your understanding.

Problem 1: You have £40 cash Turn 3. Option A: Buy basics (£6), make £18 profit. Option B: Buy premium (£14), 50% chance £28 profit, 50% chance £12 profit.

What's the EV of each option? Which should you choose?

Problem 2: Demand cards: 8 remaining, 2 show Hotel high-demand. What's probability Hotel is favorable Turn 4? If Hotel high-demand means £32 profit, medium-demand £20, what's expected profit?

Problem 3: You're at Beach (projected £14/turn × 3 turns = £42). Pivot to Hotel costs £8 transition, then £26/turn × 3 turns = £78 - £8 = £70. Worth pivoting? What if Hotel has 40% chance of competitor arriving (reducing your profit to £18/turn)?

Answers at end of article.

Connection to GCSE/A-Level Maths Curricula

GCSE Maths Links

Topics reinforced:

• Ratio and proportion (profit margins, percentages)
• Probability (demand card calculations)
• Algebraic thinking (creating formulas for profitability)
• Data handling (collecting game data, graphing results)

Activity: "Play Smoothie Wars, track your profit each turn, graph it, calculate mean/median/mode. Analyze variance. What turn was most profitable and why?"

A-Level Maths/Further Maths

Topics:

• Expected value (probability × outcomes)
• Optimization (maximizing profit functions subject to constraints)
• Decision theory (evaluating alternatives mathematically)
• Game theory basics (strategic interaction, Nash equilibrium concepts)

Activity: "Model Smoothie Wars mathematically. Create profit functions for each location. Determine optimal strategy using calculus (maximize profit function)."

Downloadable Decision Calculator

Excel/Google Sheets calculator (free download):

Inputs:

• Current cash
• Location
• Competitors at location
• Ingredients owned
• Demand card probabilities
• Turns remaining

Outputs:

• Expected profit this turn
• Recommended price
• Pivot recommendation (Yes/No + suggested location)
• Cash reserve target

Use: Training tool for understanding math, NOT for use during actual play (that's cheating/removes fun).

Download at: calculator tool page

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Practice Problem Answers

Problem 1:

• Option A EV: £18 (deterministic)
• Option B EV: (0.5 × £28) + (0.5 × £12) = £14 + £6 = £20
• Choose Option B (£20 EV > £18), IF you're risk-neutral. If risk-averse, Option A is safer.

Problem 2:

• P(Hotel high-demand) = 2/8 = 25%
• EV = (0.25 × £32) + (0.75 × £20) = £8 + £15 = £23 expected profit

Problem 3:

• Beach: £42 (deterministic)
• Hotel no competitor: £70 (as calculated)
• Hotel with competitor (40% chance): £18/turn × 3 = £54 - £8 = £46
• Hotel EV: (0.6 × £70) + (0.4 × £46) = £42 + £18.40 = £60.40
• Choose Hotel (£60.40 EV > £42 Beach)

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About the Author: James Chen applies mathematical analysis to strategy games. The team created decision frameworks and optimization models for competitive play.

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Think mathematically, win consistently. Download our Decision Calculator (Excel/Google Sheets) and Mathematical Strategy Guide (PDF). Get Smoothie Wars and apply these calculations.