Using maths to choose
Big decisions (which college, which phone plan, buying vs leasing a car, where to build a park) involve many factors. Mathematical decision-making organizes them so choices are clear, fair and explainable.
Step 1: Define options and criteria
List the options and the criteria that matter.
Example: Choosing a phone plan: cost, data, network coverage, contract length.
Step 2: Quantify
Turn criteria into numbers:
- Costs over the same period (e.g., 2 years total)
- Scores (1–10) for qualities like coverage or comfort
- Probabilities for uncertain outcomes
Tool 1: Total cost comparison
| Plan | Monthly | Setup | 24-month total |
|---|---|---|---|
| A | $45 | $0 | $1,080 |
| B | $38 | $100 | $1,012 |
| C | $55 | $0 (includes a free phone worth $300) | $1,320 − $300 = $1,020 |
Plan B is cheapest overall, even though its setup fee makes it look worse at first.
Tool 2: Weighted decision matrix
A weighted decision matrix scores each option on each criterion and multiplies by a weight for importance.
Example: Choosing between two colleges (scores out of 10).
| Criterion | Weight | College X | College Y |
|---|---|---|---|
| Program quality | 0.4 | 9 | 7 |
| Cost | 0.3 | 5 | 9 |
| Distance from home | 0.2 | 6 | 8 |
| Campus life | 0.1 | 8 | 6 |
| Weighted total | 7.1 | 7.7 |
X: 0.4(9) + 0.3(5) + 0.2(6) + 0.1(8) = 3.6 + 1.5 + 1.2 + 0.8 = 7.1
Y: 2.8 + 2.7 + 1.6 + 0.6 = 7.7
College Y scores higher with these weights.
Tool 3: Expected value
For uncertain outcomes, compare expected values.
Example: A small business can run an outdoor event (P(rain) = 0.3).
- Outdoor: profit $5,000 if dry, −$1,000 if it rains → E = 0.7(5,000) + 0.3(−1,000) = $3,200
- Indoor: guaranteed $2,500
Outdoor has the higher expected value, but more risk.
Trade-offs
Every choice involves a trade-off: lower cost vs better quality, higher reward vs higher risk. Maths makes trade-offs visible; values decide how to weigh them.
Sensitivity analysis
A sensitivity analysis asks: would the decision change if the inputs changed?
In the college example, if program quality’s weight rose to 0.6 (and cost fell to 0.1), X would score 0.6(9) + 0.1(5) + 0.2(6) + 0.1(8) = 7.9 vs Y = 0.6(7) + 0.1(9) + 0.2(8) + 0.1(6) = 7.3. The decision flips. That tells you the choice depends heavily on how much you value program quality.
Worked examples
Example 1: A gym charges $40/month or $400/year. Better deal for 12 months? Monthly = $480 → yearly plan.
Example 2: Weighted score: weights 0.5 and 0.5, scores 6 and 10? 8.
Example 3: A guaranteed $100 or a 50% chance at $250? E = $125 for the gamble; choice depends on risk tolerance.
Common mistakes
- Comparing costs over different time periods.
- Choosing weights after seeing the results to get a preferred answer.
- Ignoring risk and focusing only on expected value.
Try it yourself
- Plan A: $60/month. Plan B: $50/month + $150 setup. Cheaper over 12 months?
- Weights 0.7 (cost) and 0.3 (style). Option scores: cost 8, style 4. Weighted total?
- Option: 40% chance of $1,000, else $0. Expected value?
- Why perform a sensitivity analysis?
Answers: 1) Plan A ($720 vs $750) 2) 6.8 3) $400 4) To see whether the best choice changes when assumptions change.
Quick recap
- Define options and criteria, then quantify.
- Use total cost comparisons, a weighted decision matrix and expected value.
- Recognize trade-offs and test with sensitivity analysis.