Geometric Distribution Calculator

How it is calculated

Here p is the success probability of one trial, written as a decimal between 0 and 1, and k is the trial number. The tool counts trials starting at 1 and returns three values with five decimals: P(X = k) = (1 − p)k−1 × p, P(X ≤ k) = 1 − (1 − p)k and P(X ≥ k) = (1 − p)k−1.

The value of p must be above 0 and at most 1; a percentage such as 20 is rejected, so enter 0.2. The trial number is rounded to the nearest whole number, so 3.6 is treated as 4, and anything below 1 gives no result. The model assumes independent trials with the same p every time, and X counts the trial on which the first success happens, not the failures before it.

Common mistakes

• Entering the probability as a percent. Use 0.2, not 20; values above 1 return no result.
• Mixing up the two cumulative figures. P(X ≤ k) is the chance of succeeding within k trials; P(X ≥ k) is the chance that the first k − 1 trials all fail.
• Using the version of the distribution that counts failures before the first success. This tool counts the trial number, so k = 1 means success on the first try.
• Assuming a streak of failures makes success more likely. The trials are independent, so the chance on the next trial is still p.
• Entering a fractional k and expecting it to be used as is. It is rounded to a whole number first.