Binomial Distribution Calculator
Find P(exactly k successes) in n trials.
The binomial distribution models the number of successes in fixed independent trials.
How the Math Works
The binomial distribution calculates the probability of achieving exactly k successes in n independent trials, where each trial has the same probability p of success. The formula P = C(n,k) p^k (1-p)^(n-k) combines three key components: the binomial coefficient C(n,k) counts all possible ways to arrange k successes among n trials, p^k represents the probability of those k successes occurring, and (1-p)^(n-k) accounts for the probability of the remaining (n-k) failures. For example, if you flip a fair coin 10 times (n=10) and want exactly 5 heads (k=5) with p=0.5, the calculation determines how likely this specific outcome is across all possible sequences of flips.
Practical Applications
This calculator is essential for quality control in manufacturing, where you might test 100 products (n=100) to find the probability of exactly 3 defective items (k=3) given a known defect rate p=0.02. In clinical research, it helps determine the likelihood of observing exactly 8 successful treatments (k=8) out of 20 patients (n=20) when a new drug has a 40% success rate (p=0.4). Marketing teams use it to calculate the probability of exactly 15 customer responses (k=15) from 50 outreach attempts (n=50) when the historical response rate is 30% (p=0.3).
Day-to-Day Use
Understanding these probabilities helps you make better decisions in everyday situations. When a weather app predicts a 20% chance of rain each day this week, you can calculate the probability of exactly 2 rainy days out of 7 to plan accordingly. If you're learning a new skill with a 70% success rate per attempt, you can estimate how likely you are to master it in exactly 5 tries out of 7 attempts. Even in personal finance, if a stock has a 60% chance of gaining value daily, you can calculate the probability of it gaining value on exactly 4 out of your next 5 trading days to inform your investment timing decisions.
Worked example
10 trials, k 3, p 0.5 → 11.7%.
FAQ
Coin flips?
Use p = 0.5 — it's a special case.