Binomial Probability Calculator

Binomial Probability Formula:

P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

Where:

• n = the total number of independent trials
• k = the specific number of successes you want to find the probability for
• p = the probability of success on each individual trial (a value between 0 and 1)
• C(n,k) = the binomial coefficient, calculated as n! / (k! x (n-k)!). This counts the number of different ways to arrange k successes among n trials.

Conditions for a binomial distribution: For this formula to apply, all four conditions must be true:

1. There is a fixed number of trials (n is known in advance)
2. Each trial has only two possible outcomes (success or failure)
3. The probability of success (p) is the same for every trial
4. The trials are independent — the outcome of one does not affect the others

Practical Example: A basketball player has a 75% free throw percentage. What is the probability of making exactly 8 out of 10 free throws? n = 10, k = 8, p = 0.75 C(10,8) = 45 P(X = 8) = 45 x 0.75^8 x 0.25^2 = 45 x 0.1001 x 0.0625 = 0.2816 (28.16%)

This calculator also computes cumulative probabilities: P(X <= k), P(X >= k), P(X < k), and P(X > k). These are useful when you need to know the probability of “at most” or “at least” a certain number of successes.

When to use this calculator: Use it for quality control (probability of defective items), medical studies (treatment success rates), sports statistics, gambling odds, or any scenario with repeated yes/no outcomes.

The calculator also shows:

• Mean (expected value) = n x p
• Variance = n x p x (1-p)
• Standard deviation = square root of the variance

Tips:

• If p is close to 0 or 1, most of the probability concentrates on a few values of k
• For large n with small p, the Poisson distribution is often a better approximation
• When n is large and p is not extreme, the normal distribution can approximate the binomial