Binomial probability formula

College/University

Definition

The formula used to calculate the probability of a "Bernoulli trial" or an event with exactly two different possible outcomes, such as flipping a coin. The formula is represented as _______ where "n" is the number of trials, "k" is the number of successes, "n-k" is the number of failures, "p" is the probability of success in one trial, and "q=1-p" is the probability of failure in one trial.

Worked examples

\(P(X = 3) = \binom{5}{3}(0.6)^3(0.4)^2 = 10 \cdot 0.216 \cdot 0.16 = 0.3456\)

Probability of exactly 3 successes in 5 trials when each trial has a 60% success rate.

\(P(X = 2) = \binom{4}{2}(0.5)^2(0.5)^2 = 6 \cdot 0.25 \cdot 0.25 = 0.375\)

Probability of exactly 2 heads in 4 coin flips, where \(p = q = 0.5\).

Common mistakes

\(P(X = 3) = (0.6)^3(0.4)^2\) → \(P(X = 3) = \binom{5}{3}(0.6)^3(0.4)^2\) Forgetting the binomial coefficient undercounts the number of ways to arrange successes and failures.
\(P(X = 2) = \binom{5}{2}(0.3)^2(0.3)^3\) → \(P(X = 2) = \binom{5}{2}(0.3)^2(0.7)^3\) Use \(q = 1 - p\) for the failure probability, not \(p\) again.
Using \(n - k = 3\) as the exponent on \(q\) when \(n = 5, k = 3\) → The exponent on \(q\) is \(n - k = 2\) The number of failures is total trials minus successes: \(5 - 3 = 2\), not 3.

Where you'll use it next

You'll apply the binomial probability formula in statistics when modeling repeated independent trials, hypothesis testing, and confidence intervals, and it extends to the binomial distribution and normal approximation in advanced probability.

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Binomial Distribution: Your Gateway to Advanced Probability

Statistics

Unlock the power of binomial distribution! From basic concepts to real-world applications, master this essential statistical tool. Perfect for students and professionals alike.

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