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
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 exact 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
Use the p slider to set the probability of success on each trial. Values range from 0.01 (very rare success) to 0.99 (almost certain success), controlling how quickly you expect to see the first success.
As you increase p, the distribution becomes more concentrated at k = 1, meaning the first success is more likely to occur on the first trial. As p decreases, the distribution spreads out with longer waiting times becoming more probable.
Watch how the expected value E[X] = 1/p changes with the slider. At p = 0.5, expect 2 trials on average. At p = 0.1, expect 10 trials. The relationship is perfectly reciprocal.
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 trials. 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]=1/p, Var(X)=(1−p)/p2, σ=(1−p)/p2 and Mode=1. 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, p=0.3, the cards read 3.3333, 7.7778, 2.7889 and 1. Mode stays 1 at every p: the first trial is always the single most likely place for the first success. At p=0.10 the cards read 10.0000, 90.0000, 9.4868 and 1.
The Parameters cards above the panel repeat each value with its symbol and a one-line meaning: Success Probability (p), Failure Probability (q = 1-p) and Expected Trials (E[X]). For where the formulas come from, see Mean, Variance, and Statistics.
Reading the PMF Visualization
DemoPMF, CDF and Table tabs
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The PMF chart shows probability bars starting at k = 1 (first trial). The distribution always has its mode at k = 1, with probabilities decreasing exponentially for higher k values.
Each bar represents P(X = k), the probability that the first success occurs exactly on trial k. The height decreases by a factor of (1-p) for each subsequent trial, creating the characteristic exponentially decaying pattern.
The displayed range extends to about 30 trials by default, but the distribution theoretically continues to infinity. Probabilities become negligible beyond a few multiples of the mean 1/p.
Understanding the CDF Display
The CDF curve shows P(X ≤ k), the probability that success occurs within the first k trials. Unlike the continuous CDF for other distributions, this appears as a step function jumping at each integer value.
The CDF rises quickly when p is large, reaching values near 1 within just a few trials. For small p, the CDF rises slowly, indicating that many trials might be needed before seeing success.
At any point k, the CDF value equals 1 - (1-p)^k, providing a closed-form expression for cumulative probabilities without needing to sum individual terms.
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, p=0.3, the table runs from k=1 to k=30. Row 1 is 0.300000 and each row is 0.7 times the one before; the column stops at k=30 with P(X≤30)=0.999977, because the support never ends and the table is cut there.
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 Parameter
At p=0.3 the tallest bar is the first one, P(X=1)=0.3, and every bar after it is 0.7 times its neighbour. The mode of a geometric distribution is always k=1.
The red line sits at E[X]=1/p=3.33, well to the right of the mode.
Geometric, p = 0.3: PMF above, CDF below
The first bar is tallest at 0.3 and each after it is 0.7 times the last, so the mode is 1 while the mean is 3.33. The plotted window (k to 30) holds 0.999977 of the probability.
That gap between mode 1 and mean 3.33 is what right skew looks like: the most likely single outcome is the smallest one, but a long thin tail drags the average up.
The tool plots k=1 to 30, and those bars sum to 0.999977 rather than 1. The missing 0.0023% lives past the right edge — the support is unbounded, so any finite window falls slightly short.
The variance is (1−p)/p2=7.78, giving σ=2.79, which is almost as large as the mean itself. A distribution whose standard deviation rivals its mean is very spread out, and the CDF tab shows the consequence: it takes until about k=9 to accumulate 95% of the probability.
Computing Exact Probabilities
DemoPoint and cumulative probabilities
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Enter trial number k in the P(X = k) row of the Probability Calculator to find P(X = k) using the formula p(1-p)^(k-1). This gives the exact probability that the first success occurs on trial k.
The calculation accounts for k-1 failures (each with probability 1-p) followed by one success (probability p). For example, with p = 0.3, the probability of first success on trial 3 is 0.7² × 0.3 = 0.147.
Try different k values to see how probability decays. Each additional trial multiplies the previous probability by (1-p), creating the geometric decay that gives this distribution its name.
Calculating Survival Probabilities
Use P(X > k) to find the probability of needing more than k trials. This "survival probability" equals (1-p)^k - simply raising the failure probability to the kth power.
The memoryless property appears here: P(X > n+k | X > n) = P(X > k). If you've already failed n times, the probability of failing k more times is the same as if you were starting fresh.
P(X ≥ k) includes k itself, computed as (1-p)^(k-1). This subtle difference matters for threshold questions like "at least k trials needed."
Range Probability Calculations
DemoRange probabilities
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The P(a ≤ X ≤ b) row of the Probability Calculator finds P(a ≤ X ≤ b), the probability that first success occurs between trials a and b inclusive. This equals the difference in CDF values: F(b) - F(a-1).
Four boundary options handle edge cases: • [a, b] - Both endpoints included • (a, b) - Both endpoints excluded • [a, b) - Include a, exclude b • (a, b] - Exclude a, include b
These calculations help answer practical questions like "What's the probability I'll need between 5 and 10 attempts?"
Key Properties and Real-World Applications
The right column holds two lists. Key Properties has 5 rows, each ending in ▼: Support: X ∈ {1, 2, 3, ...}, Memoryless Property, Independent Trials, Decreasing Probabilities, Special Case of Negative Binomial. 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.
Independent Trials and Decreasing Probabilities quote the current p, so they change with the slider. Open a row, move a slider, and the numbers in the open explanation follow.
Real-World Applications, below, lists 5 situations the geometric distribution models, starting with Number of coin flips until you get heads and Number of sales calls until you make a sale. This list is fixed text and does not change with the sliders.
The geometric distribution models the number of trials needed until the first success occurs in a sequence of independent Bernoulli trials. It's the discrete analog of the exponential distribution.
Each trial must be independent with constant success probability p. The distribution counts the trial number on which success first appears, starting from trial 1 and extending theoretically to infinity.
Applications include reliability testing (trials until component failure), sales (calls until closing a deal), and quality control (inspections until finding a defect). For detailed theory and proofs, see geometric distribution theory page.
The Memoryless Property Explained
The geometric distribution is the only discrete distribution with the memoryless property: past failures don't affect future probabilities. Mathematically, P(X > n + k | X > n) = P(X > k).
This means if you've already failed n times, the probability distribution for future trials is identical to starting fresh. Each trial is a "clean slate" - previous outcomes provide no information about when success will occur.
This property arises because trials are independent with constant probability. It's what makes the geometric distribution appropriate for modeling truly random processes with no "memory" or aging effects.
Mean, Variance, and Statistics
The mean E[X] = 1/p gives the expected number of trials until first success. With p = 0.2, expect 5 trials on average. The mean is always the reciprocal of the success probability.
The variance equals (1-p)/p², measuring spread around the mean. Higher variance means more uncertainty about when success will occur. The standard deviation(1−p)/p2 provides a more interpretable measure of spread.
The mode is always 1 regardless of p - the most likely outcome is success on the first trial. However, when p < 0.5, the mean exceeds 2, showing the distribution's right skew.
Related Distributions and Calculators
The negative binomial distribution generalizes the geometric to count trials until the rth success, while geometric specifically handles r = 1. Both share the memoryless property.
The exponential distribution is the continuous version of the geometric, modeling time until an event rather than trial number. It also has the memoryless property.
Related Tools:
Negative Binomial Calculator - Trials until r successes