What is the Binomial Distribution?

A binomial distribution models the probability of obtaining a certain number of “successes” in a fixed number of trials. Each trial has exactly two possible outcomes and is independent of all other trials. The distribution is defined by two parameters n (the number of trials) and p (the probability of success in each trial).

How to use the binomial distribution calculator?

Choose an option from the drop-down list:
• Exact probability p (X = k).
• Cumulative probability p (X ≤ k).
• Range probability.
• Mean and standard deviation.
• Enter the number of trials (n)
• Enter the probability of success (p).
• Enter the number of successes (k).
• Enter range start (a) and range end (b).
• After entering the values, click “Calculate”, and in seconds, you will see the result.

Choose to Calculation Type “Exact Probability”

Find the probability of getting exactly 3 heads when tossing a coin 6 times.
n = 6, p = 0.5, k = 3
Solution:
Finding P(X = 3)
Formula: P(X = k) = C (n, k) × p ^ k × (1- p)^(n-k)
C (6, 3) = 20
P(X = 3) = 20 × 0.5^3 × 0.5^3 = 0.3125
Result:
Probability: 0.3125
Percentage: 31.25%

Choose to Calculation Type “Cumulative probability”

Number of Trials (n): 10,
Probability of Success (p): 0.30,
Range Start (a): 2,
Range End (b): 5
Result: 0.80334 (≈ 80.33%).

Choose to Calculation Type “Range probability”

Number of Trials (n): 12,
Probability of Success (p): 0.25,
Range Start (a): 3, Range End (b): 7 P(3≤X≤7) ≈0.6065 It indicates you have a 60.65% chance of scoring between 3 and 7 successes,
including both. Result 0.6065

Choose to Calculation Type “Mean and Standard Deviation”

Find the mean and standard deviation for n =10,
p = 0.6. Solution: Mean: μ = np = 10 × 0.6 = 6 Variance: σ² = np(1-p) = 10 × 0.6 × 0.4 = 2.4 Standard deviation: σ = √2.4 = 1.5492 Result: Mean = 6,
variance = 2.4, standard deviation= 1.5492

Why use the MathCalc Binomial distribution calculator?

Get Quick Results

If you are calculating exact probability, cumulative probability, or range probability by hand, it can be time-consuming, especially when dealing with large values. This free MathCalc Binomial distribution calculator gives you an accurate result in seconds.
Example: If you are finding exact probabilities of n = 6, p = 0.5, k = 3, enter your values, and the calculator gives you an accurate result (0.3125).

Reduce Human Error

Manual math can lead to minor mistakes that cost you money or points. This tool provides proven formulas to reduce errors, and your results are always right. To avoid miscalculations, use the MathCalc Binomial distribution calculator.
Example: If you are finding cumulative probabilities of n = 5, p = 0.4, k =2, enter your values, and the calculator gives you an accurate result (0.68256).

User-Friendly

This free MathCalc binomial distribution calculator calculates exact probabilities, cumulative probabilities, range probabilities, and mean and standard deviation in one calculator. This calculator is helpful for students and teachers.

Tips for Best Results

• Always double-check your input.
• Always select the correct option.

FAQ

What is the formula for binomial probability?

Formula: (P(X = k) = C (n, k) × p^k × (1- p)^(n-k)

Can this calculator handle range probabilities?

Yes, enter the start (a) and end (b) values, and it adds all probabilities between them.