Binomial Probability Calculator

Use the Binomial Probability Calculator to compute both individual and cumulative binomial probabilities with ease. Simply enter the probability of success, number of trials, and desired number of successes to instantly get results.

■ Select a probability type from the dropdown box.

■ Enter a value in each unshaded textbox.

■ Click Calculate to compute probabilities.

Probability

Probability of success on a trial

Number of trials

Number of successes (x)

Binomial probability: P(X=7)

0.1934

Cumulative probability: P(X<7)

0.6128

Cumulative probability: P(X≤7)

0.8062

Cumulative probability: P(X>7)

0.1938

Cumulative probability: P(X≥7)

0.3872

Mean (μ)

Standard deviation (σ)

1.7321

Probability distribution

Number of successes, x	P(X = x)	P(X ≤ x)
0	0.0002	0.0002

How to Use This Calculator

Probability Type: Choose what you want to calculate – exact probability or cumulative.
Success Probability (p): Enter as decimal (0.5 = 50%). This is the probability of success in each trial.
Number of Trials (n): Total number of independent events or trials.
Number of Successes (x): The number of successful outcomes you’re interested in.

Understanding Results

P(X = x): Exact probability of exactly x successes.
P(X ≤ x): Probability of x or fewer successes.
P(X ≥ x): Probability of x or more successes.
Mean (μ): Expected number of successes: μ = n × p
Standard Deviation (σ): Spread of results: σ = √(n × p × (1-p))

Real-World Examples

Quality Control: Factory defect rate 2%, batch of 100. What’s P(exactly 3 defects)?
Medical: Vaccine 95% effective, 50 vaccinated exposed. What’s P(at least 48 protected)?
Sports: Player 80% free throw rate, takes 10 shots. What’s P(makes at least 8)?
Coin Flips: Fair coin 20 flips. What’s P(exactly 12 heads)?

Probability of Getting Exactly 7 Heads in 12 Tosses

Problem:
You toss a fair coin 12 times. What is the probability of getting exactly 7 Heads?

Given:

Probability of success (Head) = 0.5
Number of trials = 12
Number of successes = 7
Calculator mode = P(X = x)

After entering the values and clicking Calculate, the Binomial Calculator shows:

👉 P(X = 7) = 0.193

This means the probability of getting exactly 7 Heads in 12 tosses is 0.193.

The calculator also provides cumulative results. For example:

👉 P(X ≤ 7) = 0.806
(Probability of getting at most 7 heads.)

Variable	Value
Probability of success (Head)	0.5
Number of trials (n)	12
Number of successes (k)	7
Goal	P(X = 7)

➤ Related