Binomial Distribution Calculator
Instantly calculate binomial probabilities with this interactive tool. Visualize distribution charts, compute exact and cumulative odds, and analyze key properties like mean and standard deviation effortlessly.
Updated December 2025
P(X = 5)
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Exact Probability
P(X ≤ 5)
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At most x
P(X ≥ 5)
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At least x
||| Probability Distribution
DISTRIBUTION PROPERTIES
MEAN (EXPECTED)
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STD DEVIATION
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Binomial Distribution Guide
Understand the mechanics of success and failure. Visualize how probability changes outcomes.
The Shape of Chance
The probability of success ($P$) dramatically changes the shape of the curve. Click below to see how a "fair" game differs from a "rigged" one.
Symmetric Distribution (Bell Shaped)
Why does this matter?
If $P$ is close to 0.5, the binomial distribution looks like a normal "Bell Curve." If $P$ is very low or very high, the data is "Skewed," meaning most outcomes bunch up on one side.
The "BINS" Assumptions
You cannot use the Binomial Calculator for just any problem. The scenario must satisfy four strict rules, remembered by the acronym BINS.
B
Binary Outcomes
There are only two possible results per trial: Success or Failure (Yes/No, Heads/Tails).
I
Independent
The result of one trial does not affect the next. (e.g., Coin flips are independent; drawing cards without replacement is not).
N
Number is Fixed
You must decide the number of trials ($n$) in advance. You cannot keep going "until" you win.
S
Same Probability
The probability of success ($p$) must remain exactly the same for every single trial.
Common Trap: Hypergeometric Distribution
If you are selecting items from a small batch without replacement, the probability changes after every pick. This violates the "I" and "S" rules. You should use the Hypergeometric Distribution instead.
The Formula Explained
How do we calculate the probability of exactly $k$ successes? The formula combines counting logic with raw probability.
The Counter
Combinations
Counts how many different ways you can arrange $k$ wins in $n$ trials (e.g., Win-Loss-Win vs. Win-Win-Loss).
The Wins
Success Probability
The probability of winning multiplied by itself $k$ times.
The Losses
Failure Probability
The probability of losing multiplied by the remaining number of trials ($n-k$).
Exact vs. Cumulative
The most confusing part of probability is the wording. "Exactly 3" is very different from "At least 3."
PROBABILITY MASS (PMF)
"Exactly 3 Successes"
We only count the specific outcome where $X$ is equal to 3. All other bars are ignored.
CUMULATIVE (CDF)
"At Most 3 Successes"
This adds up the probability of 0, 1, 2, AND 3. It answers: "What is the chance I get 3 or fewer?"
SURVIVAL FUNCTION
"At Least 3 Successes"
This looks at the upper end. It sums up the probabilities of 3, 4, and 5. It answers: "What is the chance I get 3 or more?"
EXPERT KNOWLEDGE
Common Questions
What happens if N is very large?
When the number of trials ($n$) is large, and $p$ is not too close to 0 or 1, the Binomial Distribution becomes almost identical to the Normal Distribution (Bell Curve). This is known as the Central Limit Theorem.
What is the Expected Value (Mean)?
The mean is simply $n \times p$. If you flip a coin ($p=0.5$) 100 times ($n=100$), you expect $100 \times 0.5 = 50$ heads. This is the peak of the curve.
Can I use this for defect rates?
Yes, if the defects occur independently and you are taking a fixed sample size. For example, checking 50 parts where each has a 2% chance of being bad is a classic Binomial problem.
Verified Expert
Disclaimer: This tool is for informational and educational purposes only. Calculations are based on standard formulas but may not account for unique business variables. We do not accept liability for decisions made based on these results.