Drag n and p under Parameters: n from 1 to 50 trials, p from 0.01 to 0.99. Bars, statistics and every calculator result update at once. Learn more about the parameter controls
Statistics shows Expected Value, Variance, Std Deviation and Mode for the current parameters; Show Formulas prints the rule under each card. Learn more about the statistics panel
PMF, the first tab above the chart, draws one bar per value k with height P(X=k); the red line marks E[X]. Learn more about the PMF chart
In the Probability Calculator, tap P(X = k), P(X < k), P(X ≤ k), P(X > k) or P(X ≥ k) and type k; the box below shows the probability to six decimals. Learn more about point probabilities
Tap a range tab, P(a ≤ X ≤ b), P(a < X < b), P(a ≤ X < b) or P(a < X ≤ b), and type a and b; the four differ in which endpoints count. Learn more about range probabilities
Adjust n (number of trials) using the top slider to set how many independent experiments you're running. The slider ranges from 1 to 50 trials, allowing you to model scenarios from a single coin flip to extensive quality control batches.
Set p (success probability) with the bottom slider to define the likelihood of success on each trial. Values range from 0.01 to 0.99, with 0.5 representing equally likely outcomes like fair coin flips.
Watch the visualization update in real-time as you modify parameters. The blue bars show the probability mass function (PMF), displaying the exact probability for each possible number of successes from 0 to n.
Reading the Statistics Panel
The Statistics panel sits in the left column, under Parameters. Its four cards show Expected Value, Variance and Std Deviation to four decimals, and Mode as a whole number of successes. Every card recomputes on each slider move, so the panel always describes the bars in the middle column; the red E[X] line on the chart sits at the Expected Value number.
Show Formulas prints the rule under each card: E[X]=np, Var(X)=np(1−p), σ=np(1−p) and Mode=⌊(n+1)p⌋. The button then reads Hide Formulas. The numbers are the same either way; the formulas only show where they come from.
At the opening values, n=10, p=0.5, the cards read 5.0000, 2.5000, 1.5811 and 5. Expected Value and Mode agree at 5 because p=0.5 puts the peak in the middle. Slide p to 0.20 at n=20 and the cards read 4.0000, 3.2000, 1.7889 and 4.
The Parameters cards above the panel repeat each value with its symbol and a one-line meaning: Number of Trials (n), Success Probability (p) and Failure Probability (q = 1-p). For where the formulas come from, see Understanding Distribution Parameters.
Interpreting the PMF Chart
DemoPMF, CDF and Table tabs
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The PMF (Probability Mass Function) chart displays vertical bars for each possible outcome. The height of each bar represents P(X = k), the probability of getting exactly k successes.
The distribution's shape changes dramatically with different parameters. When p = 0.5, the distribution is symmetric. As p moves toward 0 or 1, the distribution becomes increasingly skewed.
The tallest bar indicates the mode - the most likely number of successes. For n trials with probability p, the mode is approximately at np, which matches the distribution's mean.
Understanding the CDF Graph
The CDF (Cumulative Distribution Function) shows P(X ≤ k), the probability of getting k or fewer successes. Unlike the PMF's discrete bars, the CDF displays as a step function that increases from 0 to 1.
Each step in the CDF corresponds to adding one more outcome's probability. The steeper the step, the higher the probability of that specific outcome.
Use the CDF to quickly find probabilities for ranges. For example, P(X ≤ 7) can be read directly from the CDF curve at x = 7.
Using the Table Tab
Table is the third tab above the chart, after PMF and CDF. It replaces the chart with three columns, k, P(X = k) and P(X ≤ k), one row per value on the chart's axis, every value to six decimals. The rows follow the sliders: move one and the whole table recomputes.
Use it where the chart is too small to read. At the opening values, n=10, p=0.5, the table runs from k=0 to k=10. Row 5 is the largest, 0.246094, with P(X≤5)=0.623047; rows 0 and 10 hold 0.000977 each, and the last P(X≤k) is exactly 1.000000.
The P(X ≤ k) column is the CDF as numbers, and each P(X = k) entry is one bar of the PMF chart. Subtract two neighbouring P(X ≤ k) rows to recover a P(X = k) entry. For a value the table does not list, or a range of values, use the Probability Calculator below the chart.
The Explorer at Its Opening Parameters
The binomial view opens at n=10 and p=0.5, giving eleven bars from k=0 to k=10 that sum to exactly 1. The red line marks E[X]=np=5, and the tallest bar sits there too, at P(X=5)=0.2461.
Switching to the CDF tab replots the same numbers as a running total that climbs from 0.001 to 1.
Binomial, n = 10, p = 0.5: PMF above, CDF below
Eleven bars summing to exactly 1, symmetric about the peak at k = 5 where P = 0.2461 = 252/1024. Mean and mode coincide only because p = 0.5; the CDF below climbs from 0.001 to 1.
Mean and mode coincide here only because p=0.5 makes the coefficient row palindromic. Move p away from 0.5 and the mode becomes ⌊(n+1)p⌋, which parts company with np.
The variance is np(1−p)=2.5, so σ≈1.58. Almost the whole distribution lies within two standard deviations of the mean, which is why the bars at k=0 and k=10 — 1/1024 each — are barely off the axis.
Reading the two tabs together is the point of having both. The tallest PMF bar is the steepest CDF rise; a flat stretch in the CDF is a negligible PMF bar.
Calculating Point Probabilities
DemoPoint and cumulative probabilities
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Enter a value k in the P(X = k) row of the Probability Calculator to find P(X = k), the exact probability of getting k successes. The calculator uses the binomial formula: C(n,k) × p^k × (1-p)^(n-k).
The binomial coefficient C(n,k) counts the number of ways to arrange k successes among n trials. The calculator handles this computation automatically, even for large values where manual calculation would be impractical.
Try calculating P(X = 5) for n = 10, p = 0.5. You should get approximately 0.246, meaning there's about a 24.6% chance of exactly 5 successes in 10 fair coin flips.
Computing Cumulative Probabilities
The P(X ≤ k) row of the Probability Calculator computes P(X ≤ k) by summing probabilities from 0 to k. This answers questions like "What's the probability of at most k successes?"
For the complement, use P(X > k) to find the probability of more than k successes. This equals 1 - P(X ≤ k) and is useful for threshold problems.
P(X ≥ k) includes k itself in the count, while P(X > k) starts counting from k+1. Understanding this distinction prevents off-by-one errors in probability calculations.
Range Probability Calculations
DemoRange probabilities
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The Range Probability section calculates P(a ≤ X ≤ b), the probability that X falls between two values inclusive. Enter lower bound a and upper bound b to see how much probability mass lies in that interval.
Four range options handle boundary conditions differently: • [a, b] includes both endpoints • (a, b) excludes both endpoints • [a, b) includes a, excludes b • (a, b] excludes a, includes b
For discrete distributions, these distinctions matter. P(3 ≤ X ≤ 7) includes outcomes 3, 4, 5, 6, and 7, while P(3 < X < 7) includes only 4, 5, and 6.
Key Properties and Real-World Applications
The right column holds two lists. Key Properties has 5 rows, each ending in ▼: Support: X ∈ {0, 1, 2, ..., 10}, Independent Trials, Binary Outcomes, Fixed Number of Trials, Approximates Normal Distribution. Row titles here are those at the opening values. Tap a row to open its explanation; tap another and the first closes, so one explanation is open at a time; tap the open row to close it.
Every explanation quotes the current n or p: the support row reads up to n, and Approximates Normal Distribution gives the matching mean and standard deviation (at n=10, p=0.5: 5.00 and 1.58). Open a row, move a slider, and the numbers in the open explanation follow.
Real-World Applications, below, lists 5 situations the binomial distribution models, starting with Number of heads in 10 coin flips and Number of defective items in a batch of products. This list is fixed text and does not change with the sliders.
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It's one of the most fundamental discrete probability distributions.
The distribution requires four conditions: (1) fixed number of trials n, (2) each trial has two outcomes (success/failure), (3) constant success probability p, and (4) trials are independent.
Common applications include quality control (number of defective items), clinical trials (number of patients responding to treatment), and survey sampling (number of favorable responses). For comprehensive theory and formulas, see binomial distribution theory page.
Understanding Distribution Parameters
The distribution has two parameters: n determines the number of trials, while p sets the success probability for each trial. Together, they completely specify the distribution's behavior.
The mean (expected value) equals np - if you have 100 trials with p = 0.3, expect about 30 successes on average. The variance equals np(1-p), measuring spread around the mean.
The standard deviationnp(1−p) indicates typical deviation from the mean. When p = 0.5, variance is maximized at n/4, creating the widest spread for a given n.
Normal Approximation Rule
When n is large and p isn't too close to 0 or 1, the binomial distribution approximates a normal distribution. The rule of thumb: use normal approximation when both np ≥ 5 and n(1-p) ≥ 5.
The approximating normal distribution has mean μ = np and standard deviation σ = np(1−p). Apply a continuity correction by adding/subtracting 0.5 to discrete values for better accuracy.
This approximation simplifies calculations for large n where computing exact binomial probabilities becomes computationally expensive. For detailed guidance on when to use this approximation, see normal approximation to binomial.
Related Distributions and Tools
The geometric distribution is related - it counts trials until the first success, while binomial counts successes in n trials. The negative binomial distribution generalizes geometric to r successes.
For large n and small p, the Poisson distribution approximates the binomial with λ = np. This is useful when modeling rare events in large populations.