geometpdf & geometcdf Calculator

Evaluate the TI-84 geometric functions online. geometpdf(p, x) is the probability that the first success happens on trial x; geometcdf(p, x) is the probability it happens on or before trial x.

Using a textbook that counts failures before the first success instead? The geometric distribution calculator supports both conventions.

Part of the TI-84 statistics functions guide, which shows when to use geometpdf and geometcdf and every other DISTR-menu function.

geometpdf vs. geometcdf

A geometric setting repeats independent trials with the same success probability p until the first success. X counts the trial on which that success arrives, so X = 1, 2, 3, …

geometpdf(p, x) = (1 − p)^(x − 1) · p = P(X = x)

geometcdf(p, x) = 1 − (1 − p)^x = P(X ≤ x)

P(X > x) = (1 − p)^x (the first x trials all fail)

Worked Example: First Free Throw Made

A beginner makes 20% of free throws. What is the probability the first make comes on the 4th attempt, and within the first 4?

  1. geometpdf(0.2, 4) = 0.8³ × 0.2 = 0.512 × 0.2 = 0.1024.
  2. geometcdf(0.2, 4) = 1 − 0.8⁴ = 1 − 0.4096 = 0.5904.
  3. Expected number of attempts until the first make: 1 / 0.2 = 5.

The chance of needing more than 4 attempts is 0.8⁴ = 0.4096 — even with an average of 5 attempts, missing four in a row is common. The at-least-one calculator answers the same question from the other side.

Where to Find geometpdf on a TI-84

  1. Press 2nd then VARS (DISTR).
  2. Scroll to E:geometpdf( or F:geometcdf(.
  3. Enter p and the x value, then select Paste and press ENTER.

Excel has no geometric function, but =(1-p)^(x-1)*p and =1-(1-p)^x reproduce both.

Frequently Asked Questions

What is the difference between geometric and binomial?

Binomial fixes the number of trials and counts successes. Geometric fixes the number of successes at one and counts the trials needed to get it. Use geometpdf when the question asks 'on which trial' or 'how many tries until'.

Does the TI-84 count failures or trials?

Trials. geometpdf(p, x) on the TI-84 is the probability that the first success happens on trial x, so x starts at 1. Some textbooks define X as the number of failures before the first success; in that case add 1 to x before using the calculator.

How do I find 'at least x trials' with geometcdf?

P(X ≥ x) = 1 − geometcdf(p, x − 1), which simplifies to (1 − p)^(x − 1): the first x − 1 trials must all fail.

What is the expected value of a geometric distribution?

1/p trials, with variance (1 − p)/p². With p = 0.2 you expect 5 trials on average, but the distribution is right-skewed: the most likely single outcome is always x = 1.

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