The binomial distribution
When you count successes in a fixed number of independent yes-or-no trials, the binomial distribution gives the exact probabilities.
Binomial settings
A binomial setting has four features, remembered as BINS: Binary outcomes (success or failure), Independent trials, a fixed Number of trials \(n\), and the Same probability of success \(p\) on each trial. Counting heads in 10 coin flips or defective items in a sample of 20 (from a large batch) fit this model.
The binomial formula
The probability of exactly \(k\) successes is \(\binom{n}{k}p^k(1-p)^{n-k}\). The combination counts which trials are successes; the powers give the probability of each such arrangement. For 3 heads in 5 fair flips: \(\binom{5}{3}(0.5)^5 = \frac{10}{32} = 0.3125\).
P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}Mean and standard deviation
For a binomial variable, \(\mu = np\) and \(\sigma = \sqrt{np(1-p)}\). With 100 flips of a fair coin, expect 50 heads with standard deviation 5. These formulas save summing over every possible value.
\mu = np,\quad \sigma = \sqrt{np(1-p)}Worked example: a multiple-choice guesser
A student guesses on 10 questions with 4 choices each, so \(p = 0.25\). The expected number correct is \(2.5\). The probability of passing with at least 6 correct is about \(0.02\), computed by adding \(P(6)\) through \(P(10)\). Guessing is a poor strategy, and the binomial model shows exactly how poor.
Shape of binomial distributions
When \(p = 0.5\), the distribution is symmetric. When \(p\) is small, it is skewed right; when \(p\) is large, skewed left. As \(n\) grows, the distribution becomes more bell-shaped. If \(np \ge 10\) and \(n(1-p) \ge 10\), a normal distribution approximates it well.
The geometric distribution
If you repeat independent trials until the first success, the number of trials needed is geometric: \(P(X = k) = (1-p)^{k-1}p\), with mean \(\frac{1}{p}\). Rolling a die until the first six takes 6 rolls on average.
The Poisson distribution
For counts of events in a fixed interval, such as emails per hour or accidents per month, the Poisson distribution with mean \(\lambda\) gives \(P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}\). Its variance also equals \(\lambda\). It approximates the binomial when \(n\) is large and \(p\) is small.
Common mistakes
Using the binomial formula when trials are not independent or \(p\) changes. Forgetting the combination factor. Confusing 'at least' with 'exactly'. Using \(np\) for the standard deviation instead of \(\sqrt{np(1-p)}\).
Key takeaways
Check BINS before using the binomial model. \(P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}\), with mean \(np\) and standard deviation \(\sqrt{np(1-p)}\). Geometric counts trials until the first success; Poisson counts events in an interval.
Flashcard
Binary, Independent, Number fixed, Same p
Check your understanding
A fair coin is flipped 20 times. What is the expected number of heads and its standard deviation?