What is the binomial distribution formula?+
The formula is P(X = k) = C(n,k) x p^k x (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the success probability per trial, and C(n,k) = n! / (k!(n-k)!) is the binomial coefficient. This gives the probability of getting exactly k successes in n independent trials.
What are the four conditions for the binomial distribution?+
The four conditions are: (1) a fixed number of trials n; (2) each trial is independent (the outcome of one trial does not affect others); (3) each trial has only two possible outcomes (success and failure); and (4) the probability of success p is constant across all trials. If any condition fails, the binomial model does not apply.
What is the mean and standard deviation of the binomial distribution?+
The mean is mu = np. The variance is sigma^2 = np(1-p). The standard deviation is sigma = sqrt(np(1-p)). For example, with n = 50 and p = 0.4, mean = 20, variance = 12, standard deviation = 3.464. These formulas hold for any valid n and p.
What is the difference between P(X=k), P(X≤k), and P(X≥k)?+
P(X=k) is the exact probability of getting exactly k successes (the PMF value). P(X≤k) is the cumulative probability of getting k or fewer successes (the CDF value). P(X≥k) is the upper tail probability of getting k or more successes. Note that P(X≤k) + P(X≥k) - P(X=k) = 1, since P(X=k) is counted in both cumulative terms.
When can I use the normal approximation to the binomial?+
The normal approximation N(np, np(1-p)) is reliable when both np > 5 and n(1-p) > 5. For example, with n = 100 and p = 0.3, np = 30 and n(1-p) = 70, both well above 5. For better accuracy, apply a continuity correction: P(X = k) is approximated by P(k - 0.5 < Z < k + 0.5) where Z is the standard normal variable.
What is the difference between the binomial and Poisson distributions?+
The binomial distribution counts successes in a fixed number of trials n with constant probability p. The Poisson distribution counts events in a fixed time or space interval, with no upper bound on the count. As n becomes large and p becomes small while np = lambda stays constant, the binomial converges to the Poisson distribution with parameter lambda = np.
How do I calculate cumulative binomial probability by hand?+
P(X ≤ k) is the sum of P(X = 0) + P(X = 1) + ... + P(X = k), where each term uses C(n,i) x p^i x (1-p)^(n-i). For large n this becomes tedious, which is why calculators and statistical tables are used. The Distribution Table mode above shows all values instantly.
Can the binomial distribution be used for sampling without replacement?+
Technically, sampling without replacement violates the independence requirement. However, if the population is large relative to the sample size (population N >= 20 x sample size n), the binomial is a good approximation. When independence is truly violated, the hypergeometric distribution is the correct model to use.
What is the mode of the binomial distribution?+
The mode is the most probable value of X. For a binomial(n, p) distribution, the mode is floor((n+1)p) when (n+1)p is not an integer. When (n+1)p is an integer, both (n+1)p and (n+1)p - 1 are modes. The Distribution Table mode above makes it easy to identify the maximum probability row visually.
How does a binomial distribution change shape for different p values?+
For p = 0.5 the distribution is perfectly symmetric around np. For p < 0.5 the distribution is right-skewed, with most probability weight at low k values. For p > 0.5 it is left-skewed. As n increases, the distribution becomes more symmetric and increasingly resembles a normal curve, regardless of p, by the Central Limit Theorem.
What is the complement rule for binomial probability?+
The complement rule states P(X ≥ k) = 1 - P(X ≤ k-1). This is useful when computing the probability of at least k successes is easier via the complement. For example, P(at least 1 success) = 1 - P(X = 0) = 1 - (1-p)^n, which is far simpler than summing P(X=1) through P(X=n).