Lottery Odds & Expected Value Demo
Explore how combinations determine lottery odds and learn why expected value is typically negative. Educational tool only - not gambling advice.
Lottery Odds & Expected Value Demo
Set up a simple lottery-style game to see how unlikely jackpots really are and learn how expected value works. This is an educational demo only - not gambling or financial advice.
Configure Game
Choose pool size, numbers drawn, and prize tiers
See the Odds
Discover how combinations determine probabilities
Learn Expected Value
Understand why lotteries have negative EV
This tool is for educational purposes only. It does not provide gambling, financial, or investment advice.
Understanding Lottery Odds and Expected Value
Learn how probability and expected value apply to lottery-style games
How Lottery Combinations Work
In a simple lottery, you pick K numbers from a pool of N numbers. The number of possible combinations is calculated using the combination formula:
C(N, K) = N! / (K! × (N-K)!)
For a 6-from-49 lottery: C(49, 6) = 13,983,816 possible combinations. This means the odds of picking the exact winning combination are 1 in 13,983,816 - roughly 1 in 14 million.
Calculating Odds for Partial Matches
To find the probability of matching exactly r numbers out of K:
P(match r) = C(K, r) × C(N-K, K-r) / C(N, K)
This accounts for:
- C(K, r): Ways to choose r matching numbers from the K drawn
- C(N-K, K-r): Ways to choose the remaining (K-r) non-matching numbers from the (N-K) numbers not drawn
- C(N, K): Total possible combinations (in the denominator)
What is Expected Value?
Expected value (EV) represents the average outcome over many repetitions. For a lottery ticket:
EV = Σ(Probability_i × Prize_i) - Ticket Cost
If EV is negative (which it almost always is for lotteries), you lose money on average over many plays. The "house" keeps the difference between ticket sales and prizes paid.
Why Lotteries Always Have Negative Expected Value
- • Business model: Lotteries are designed to generate revenue. Total prizes are always less than total ticket sales.
- • Taxes: Winners often pay significant taxes on large prizes (not modeled here).
- • Shared jackpots: Multiple winners split the prize, reducing individual expected value.
- • Annuity vs lump sum: Advertised jackpots are often annuity values; lump sum payouts are smaller.
Example: Simple 6-from-49 Lottery
Setup: Pick 6 numbers from 49. Ticket costs $2.
Jackpot (match 6): $10,000,000
Odds of jackpot: 1 in 13,983,816
EV from jackpot alone: $10,000,000 ÷ 13,983,816 ≈ $0.715
EV per ticket: ~$0.715 (from jackpot) + small amounts from lower tiers - $2 ticket cost = negative
Important Disclaimer
- • This tool is for educational purposes only
- • It does not provide gambling, financial, or investment advice
- • Real lottery rules are more complex than this simplified model
- • We do not encourage playing the lottery or any gambling
- • The best strategy is always proper financial planning, not gambling
Frequently Asked Questions
Common questions about lottery odds and expected value
No. This tool is for educational probability learning only. It demonstrates how combinations determine lottery odds and how expected value works. It does not provide gambling, financial, or investment advice. We do not encourage playing the lottery or any other form of gambling.
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Fun & ExplorersA better alternative to gambling: Instead of playing the lottery, consider exploring our investment calculators to see how saving and investing consistently can build real wealth over time with a positive expected return.