## The Binomial Setting
The binomial distribution models the number of successes in n independent trials, where each trial has probability p of success. In betting, this applies whenever you have a series of yes/no outcomes.
## Examples in Betting
**Win rate over N bets:** If your true win rate is 48% (even-money betting with slight edge), what is the probability of winning exactly 52 out of 100 bets?
P(X = 52) = C(100,52) × 0.48^52 × 0.52^48 ≈ 5.4%
**Clean sheet probability over a season:** A team keeps clean sheets in 32% of matches. What is the probability they keep 10+ clean sheets in a 38-match season?
P(X ≥ 10) = Σ P(X = k) for k = 10 to 38 ≈ 72%
## Calculating Binomial Probabilities
Binomial CDF gives cumulative probabilities: P(X ≤ k) = Σ_{i=0}^{k} C(n,i) × p^i × (1−p)^(n−i)
Tools: Excel BINOM.DIST(), Python scipy.stats.binom, R pbinom()
## The Normal Approximation
For large n (n > 30), the binomial distribution is well approximated by the normal distribution with:
μ = np
σ = √(np(1−p))
For 500 bets with p = 0.50: μ = 250, σ = √125 ≈ 11.2
P(winning at least 260 bets) = P(Z > (260−250)/11.2) = P(Z > 0.89) ≈ 18.7%
## Practical Applications
- Calculating the probability of a specific winning or losing run
- Determining whether your observed win rate is statistically significant
- Setting realistic expectations for any binary-outcome market
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