Maths⏱ 5 min read
How to Calculate Expected Value and Use It to Make Better Decisions
Expected value is the foundation of rational decision-making under uncertainty. Here is how to calculate it, why people systematically get it wrong, and how to use it in real decisions.
Expected value (EV) is the probability-weighted average of all possible outcomes. It is the single most important concept in probability, used in gambling, insurance, finance, and everyday decision-making.
The Expected Value Formula
EV = sum of (Probability x Outcome) for all possible outcomes
Simple example: coin flip bet
You win £15 if heads, lose £10 if tails
P(heads) = 0.5, P(tails) = 0.5
EV = (0.5 x +£15) + (0.5 x -£10)
= £7.50 + (-£5.00)
= +£2.50
Positive EV (+£2.50): you should take this bet.
Negative EV: you should decline.
Zero EV: mathematically fair (neither side has advantage)
Expected Value in Gambling
UK National Lottery (main draw):
Ticket cost: £2.00
Probability of jackpot: 1 in 45,057,474
Average jackpot: ~£5,000,000
EV from jackpot alone:
(1/45,057,474) x £5,000,000 = £0.111
Including other prizes:
Match 5 + bonus ball (1 in 7,509,579): ~£1,000,000 expected share
Match 5 (1 in 144,415): ~£1,750
Match 4 (1 in 2,180): £140
Match 3 (1 in 97): £30
Total EV of all prizes: approximately £0.45-£0.55 per £2 ticket
EV = -£1.45 to -£1.55 per ticket (return of 22-28p per £1 spent)
The lottery has approximately 25% return on investment.
The remaining 75% funds prizes, operating costs, and good causes.
Expected Value in Insurance Decisions
Home contents insurance:
Annual premium: £180
Probability of a claim in any year: approximately 3%
Average claim value: £2,500
EV of insurance company payout: 0.03 x £2,500 = £75
Expected VALUE to you per year: £75 - £180 = -£105
Negative EV -- insurance is mathematically expensive.
Yet it is still rational to buy because:
- Risk aversion: losing £2,500 hurts more than gaining £2,500 helps
- Large losses are not recoverable from monthly budget
- Peace of mind has value
Key insight: insurance is rational for large, unaffordable losses.
For small losses (extended warranties, phone screen insurance):
EV is usually so negative that self-insurance is almost always better.
Multi-Outcome Expected Value
Business decision: launch a new product
Outcome A: Success (P=0.25) -- profit £200,000
Outcome B: Moderate success (P=0.40) -- profit £40,000
Outcome C: Break even (P=0.20) -- £0
Outcome D: Failure (P=0.15) -- loss £80,000
EV = (0.25 x 200,000) + (0.40 x 40,000) + (0.20 x 0) + (0.15 x -80,000)
= 50,000 + 16,000 + 0 + (-12,000)
= +£54,000
Positive EV: rational to proceed if the probabilities are accurate.
The challenge: estimating probabilities is the hard part.
Most people overestimate success probability (optimism bias)
and underestimate failure probability (planning fallacy).