Example 1
\(\text{Coin flips: } n=10, p=0.5, k=7 \rightarrow P(7 \text{ heads}) = 11.72\%\)
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Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It applies when:
- There are a fixed number of trials (n)
- Each trial has only two possible outcomes (success or failure)
- The probability of success (p) is the same for each trial
- The trials are independent
Key Formulas:
- P(X = k) = C(n, k) × p^k × (1-p)^(n-k)
- Mean: μ = n × p
- Variance: σ² = n × p × (1-p)
- Standard Deviation: σ = √(n × p × (1-p))
Common applications include quality control, clinical trials, survey sampling, and any scenario involving repeated trials with binary outcomes.
\(P(X = k) = C(n, k) \times p^k \times (1-p)^{n-k}, \mu = np, \sigma^2 = np(1-p)\)
Worked Examples
Example 2
\(\text{Quality control: } n=20, p=0.1, k=2 \rightarrow P(2 \text{ defects}) = 28.52\%\)